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:heavy_check_mark: verify/ds/persistent_cow.test.cpp

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Code

#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"

#include "../../acted_monoid/range_add_range_sum.hpp"
#include "../../ds/dsu/persistent_dsu.hpp"
#include "../../ds/dsu/persistent_potentialized_dsu.hpp"
#include "../../ds/bst/persistent_ordered_multiset.hpp"
#include "../../ds/dynamic_array/persistent_dynamic_array.hpp"
#include "../../ds/dynamic_array/persistent_dynamic_lazy_monoid_array.hpp"
#include "../../ds/dynamic_array/persistent_dynamic_monoid_array.hpp"
#include "../../ds/segtree/persistent_dual_segtree.hpp"
#include "../../ds/segtree/persistent_dynamic_dual_segtree.hpp"
#include "../../ds/segtree/persistent_dynamic_lazy_segtree.hpp"
#include "../../ds/segtree/persistent_dynamic_segtree.hpp"
#include "../../ds/segtree/persistent_lazy_segtree.hpp"
#include "../../ds/segtree/persistent_segtree.hpp"
#include "../../ds/segtree/persistent_segtree_beats.hpp"
#include "../../monoid/add.hpp"
#include "../../utilities/fast_io.hpp"

#include <cassert>
#include <cstdint>
#include <utility>
#include <vector>

namespace {

using Add = m1une::monoid::Add<long long>;
using RangeAdd = m1une::acted_monoid::RangeAddRangeSum<long long>;

struct RangeAddBeats : RangeAdd {
    static bool can_apply(const operator_type&, const value_type&) {
        return true;
    }
};

template <class Structure, class Update, class Get>
void check_copy_move_and_reuse(Structure base, Update update, Get get) {
    const std::size_t base_nodes = base.node_count();
    Structure current = base;
    Structure sibling = current;

    update(current, 0);
    assert(get(base) == 0);
    assert(get(sibling) == 0);
    assert(get(current) == 1);

    update(sibling, 9);
    assert(get(base) == 0);
    assert(get(current) == 1);
    assert(get(sibling) == 10);

    const std::size_t after_first_write = current.node_count();
    update(current, 1);
    assert(get(current) == 2);
    assert(current.node_count() == after_first_write);

    Structure snapshot = current;
    update(current, 2);
    assert(get(snapshot) == 2);
    assert(get(current) == 3);
    const std::size_t after_reshare = current.node_count();
    update(current, 3);
    assert(get(snapshot) == 2);
    assert(get(current) == 4);
    assert(current.node_count() == after_reshare);

    Structure moved(std::move(current));
    update(moved, 4);
    assert(get(moved) == 5);
    Structure assigned = base;
    assigned = std::move(moved);
    update(assigned, 5);
    assert(get(assigned) == 6);
    assert(get(base) == 0);

    sibling.release();
    snapshot.release();
    assigned.release();
    assert(base.node_count() == base_nodes);

    Structure reused = base;
    update(reused, 6);
    assert(get(reused) == 7);
    reused.release();
    assert(base.node_count() == base_nodes);
}

void test_segment_trees() {
    using Seg = m1une::ds::PersistentSegtree<Add>;
    check_copy_move_and_reuse(
        Seg(std::vector<long long>(8, 0)),
        [](Seg& seg, int step) { seg.set_inplace(3, step + 1); },
        [](const Seg& seg) { return seg.get(3); }
    );

    using Lazy = m1une::ds::PersistentLazySegtree<RangeAdd>;
    check_copy_move_and_reuse(
        Lazy(std::vector<long long>(8, 0)),
        [](Lazy& seg, int step) { seg.set_inplace(3, RangeAdd::make(step + 1)); },
        [](const Lazy& seg) { return seg.get(3).sum; }
    );

    Lazy base(std::vector<long long>(8, 0));
    Lazy current = base;
    current.apply_inplace(0, 8, 10);
    const std::size_t after_full_cover = current.node_count();
    current.apply_inplace(0, 8, 1);
    assert(current.node_count() == after_full_cover);
    Lazy snapshot = current;
    current.apply_inplace(2, 6, 5);
    current.set_inplace(3, RangeAdd::make(100));
    assert(base.all_prod().sum == 0);
    assert(snapshot.all_prod().sum == 88);
    assert(snapshot.get(3).sum == 11);
    assert(current.get(2).sum == 16);
    assert(current.get(3).sum == 100);

    Lazy persistent = base;
    Lazy inplace = base;
    std::uint64_t state = 1;
    for (int operation = 0; operation < 500; ++operation) {
        state = state * 6364136223846793005ULL + 1;
        int left = int(state % 9);
        state = state * 6364136223846793005ULL + 1;
        int right = int(state % 9);
        if (left > right) std::swap(left, right);
        long long add = static_cast<long long>(operation % 13) - 6;
        persistent = persistent.apply(left, right, add);
        inplace.apply_inplace(left, right, add);
        if (operation % 17 == 0) {
            int index = operation % 8;
            auto value = RangeAdd::make(operation);
            persistent = persistent.set(index, value);
            inplace.set_inplace(index, value);
        }
        auto inplace_values = inplace.to_vector();
        auto persistent_values = persistent.to_vector();
        assert(inplace_values.size() == persistent_values.size());
        for (int i = 0; i < int(inplace_values.size()); ++i) {
            assert(inplace_values[i].sum == persistent_values[i].sum);
        }
    }
    assert(base.all_prod().sum == 0);

    using Dual = m1une::ds::PersistentDualSegtree<Add>;
    Dual dual_base(std::vector<long long>(8, 0));
    Dual dual = dual_base;
    dual.apply_inplace(1, 7, 4);
    dual.apply_inplace(3, 6);
    dual.set_inplace(5, 20);
    assert(dual_base.get(5) == 0);
    assert(dual.get(3) == 10);
    assert(dual.get(5) == 20);

    using Beats = m1une::ds::PersistentSegtreeBeats<RangeAddBeats>;
    Beats beats_base(std::vector<long long>(8, 0));
    Beats beats = beats_base;
    beats.apply_inplace(1, 7, 4);
    const std::size_t beats_nodes = beats.node_count();
    beats.apply_inplace(1, 7, 3);
    assert(beats.node_count() == beats_nodes);
    Beats beats_copy = beats;
    beats.set_inplace(3, RangeAddBeats::make(50));
    assert(beats_base.all_prod().sum == 0);
    assert(beats_copy.get(3).sum == 7);
    assert(beats.get(3).sum == 50);
}

void test_dynamic_segment_trees() {
    using Seg = m1une::ds::PersistentDynamicSegtree<Add>;
    Seg base(-100, 100);
    Seg current = base;
    current.set_inplace(17, 1);
    const std::size_t nodes = current.node_count();
    current.set_inplace(17, 2);
    assert(current.node_count() == nodes);
    Seg copy = current;
    current.set_inplace(17, 3);
    assert(base.get(17) == 0);
    assert(copy.get(17) == 2);
    assert(current.get(17) == 3);

    using Lazy = m1une::ds::PersistentDynamicLazySegtree<RangeAdd>;
    Lazy lazy_base(-64, 64, RangeAdd::make(0));
    Lazy lazy = lazy_base;
    lazy.apply_inplace(-20, 30, 5);
    Lazy lazy_copy = lazy;
    lazy.apply_inplace(-5, 10, 7);
    lazy.set_inplace(0, RangeAdd::make(100));
    assert(lazy_base.get(0).sum == 0);
    assert(lazy_copy.get(0).sum == 5);
    assert(lazy.get(-6).sum == 5);
    assert(lazy.get(-5).sum == 12);
    assert(lazy.get(0).sum == 100);

    using Dual = m1une::ds::PersistentDynamicDualSegtree<Add>;
    Dual dual_base(-64, 64, 0);
    Dual dual = dual_base;
    dual.apply_inplace(-20, 30, 5);
    Dual dual_copy = dual;
    dual.apply_inplace(-5, 10, 7);
    dual.set_inplace(0, 100);
    assert(dual_base.get(0) == 0);
    assert(dual_copy.get(0) == 5);
    assert(dual.get(-5) == 12);
    assert(dual.get(0) == 100);
}

void test_dynamic_arrays() {
    using Array = m1une::ds::PersistentDynamicArray<int>;
    Array base = {0, 1, 2, 3, 4, 5};
    Array reversed = base.reverse(1, 6);
    Array copy = reversed;
    reversed.set_inplace(1, 20);
    const std::size_t nodes = reversed.node_count();
    reversed.set_inplace(1, 21);
    assert(reversed.node_count() == nodes);
    assert(base.to_vector() == (std::vector<int>{0, 1, 2, 3, 4, 5}));
    assert(copy.to_vector() == (std::vector<int>{0, 5, 4, 3, 2, 1}));
    assert(reversed.to_vector() == (std::vector<int>{0, 21, 4, 3, 2, 1}));

    using MonoidArray = m1une::ds::PersistentDynamicMonoidArray<Add>;
    MonoidArray monoid(std::vector<long long>{1, 2, 3, 4});
    MonoidArray monoid_copy = monoid;
    monoid.set_inplace(2, 30);
    assert(monoid_copy.all_prod() == 10);
    assert(monoid.all_prod() == 37);

    using LazyArray = m1une::ds::PersistentDynamicLazyMonoidArray<RangeAdd>;
    LazyArray lazy_base(std::vector<long long>{1, 2, 3, 4, 5, 6});
    LazyArray lazy = lazy_base.apply(1, 6, 10).reverse(0, 5);
    LazyArray lazy_copy = lazy;
    lazy.apply_inplace(1, 5, 7);
    lazy.set_inplace(2, RangeAdd::make(100));
    assert(lazy_base.all_prod().sum == 21);
    std::vector<long long> copy_expected = {15, 14, 13, 12, 1, 16};
    auto copy_values = lazy_copy.to_vector();
    for (int i = 0; i < int(copy_expected.size()); ++i) {
        assert(copy_values[i].sum == copy_expected[i]);
    }
    std::vector<long long> expected = {15, 21, 100, 19, 8, 16};
    auto values = lazy.to_vector();
    for (int i = 0; i < int(expected.size()); ++i) assert(values[i].sum == expected[i]);
    assert(lazy.all_prod().sum == 179);
}

void test_dsu() {
    m1une::ds::PersistentDsu base(8);
    auto current = base;
    assert(current.merge_inplace(0, 1));
    const std::size_t after_merge = current.node_count();
    assert(!current.merge_inplace(0, 1));
    assert(current.node_count() == after_merge);
    auto copy = current;
    assert(current.merge_inplace(1, 2));
    assert(!base.same(0, 1));
    assert(!copy.same(0, 2));
    assert(current.same(0, 2));

    using PotentialDsu = m1une::ds::PersistentPotentializedDsu<Add>;
    PotentialDsu potential_base(8);
    PotentialDsu potential = potential_base;
    assert(potential.merge_inplace(0, 1, 3));
    PotentialDsu potential_copy = potential;
    assert(potential.merge_inplace(1, 2, 4));
    assert(!potential.merge_inplace(0, 2, 8));
    assert(potential.diff(0, 2) == 7);
    assert(!potential_base.same(0, 1));
    assert(!potential_copy.same(0, 2));
}

void test_ordered_multiset() {
    using Multiset = m1une::ds::PersistentOrderedMultiset<int>;
    Multiset base = {1, 2, 2, 3};
    Multiset current = base;
    current.insert_inplace(2, 3);
    const std::size_t after_first_write = current.node_count();
    current.insert_inplace(2);
    assert(current.node_count() == after_first_write);
    Multiset copy = current;
    assert(current.erase_inplace(2));
    assert(base.count(2) == 2);
    assert(copy.count(2) == 6);
    assert(current.count(2) == 5);
    current.insert_inplace(4);
    assert(!base.contains(4));
    assert(current.contains(4));
    assert(current.erase_all_inplace(4));
    assert(!current.contains(4));
    assert(!current.erase_inplace(9));
}

}  // namespace

int main() {
    test_segment_trees();
    test_dynamic_segment_trees();
    test_dynamic_arrays();
    test_dsu();
    test_ordered_multiset();

    m1une::utilities::FastInput input;
    m1une::utilities::FastOutput output;
    long long a, b;
    input >> a >> b;
    output << a + b << '\n';
}
#line 1 "verify/ds/persistent_cow.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"

#line 1 "acted_monoid/range_add_range_sum.hpp"



namespace m1une {
namespace acted_monoid {

template <typename T>
struct RangeAddRangeSumNode {
    T sum;
    long long size;
};

template <typename T>
struct RangeAddRangeSum {
    using value_type = RangeAddRangeSumNode<T>;
    using operator_type = T;
    static constexpr bool commutative = true;
    static constexpr bool operator_commutative = true;

    // Value Monoid (Sum)
    static constexpr value_type id() {
        return {T(0), 0};
    }
    static constexpr value_type op(const value_type& a, const value_type& b) {
        return {a.sum + b.sum, a.size + b.size};
    }
    static constexpr value_type inv(const value_type& x) {
        return {-x.sum, -x.size};
    }

    // Operator Monoid (Add)
    static constexpr operator_type op_id() {
        return 0;
    }
    static constexpr operator_type op_comp(const operator_type& f, const operator_type& g) {
        return f + g;
    }

    // Mapping (sum + f * size)
    static constexpr value_type mapping(const operator_type& f, const value_type& x) {
        return {x.sum + f * x.size, x.size};
    }

    // Helper for initializing a leaf node
    static constexpr value_type make(const T& val) {
        return {val, 1};
    }
};

}  // namespace acted_monoid
}  // namespace m1une


#line 1 "ds/dsu/persistent_dsu.hpp"



#include <algorithm>
#include <cassert>
#include <cstddef>
#include <memory>
#include <utility>
#include <vector>

#line 1 "ds/detail/persistent_binary_node_pool.hpp"



#line 6 "ds/detail/persistent_binary_node_pool.hpp"
#include <deque>
#include <limits>
#include <optional>
#line 11 "ds/detail/persistent_binary_node_pool.hpp"

namespace m1une {
namespace ds {
namespace detail {

// Node must have integer `l` and `r` members. New nodes initially have no
// owner; discard_unreferenced() removes temporary path-copy nodes after the
// result roots have been retained.
template <class Node, int null_node = -1>
struct PersistentBinaryNodePool {
   private:
    std::deque<std::optional<Node>> _nodes;
    std::vector<int> _references;
    std::vector<int> _next_free;
    std::vector<int> _unowned;
    int _first_free = -1;
    std::size_t _live_nodes = 0;

    void release_zero(int node) {
        assert(node != null_node && _nodes[node].has_value());
        int left = (*_nodes[node]).l;
        int right = (*_nodes[node]).r;
        _nodes[node].reset();
        _next_free[node] = _first_free;
        _first_free = node;
        --_live_nodes;
        if (left != null_node && --_references[left] == 0) release_zero(left);
        if (right != null_node && --_references[right] == 0) release_zero(right);
    }

   public:
    PersistentBinaryNodePool() {
        if constexpr (null_node == 0) {
            _nodes.emplace_back();
            _references.push_back(0);
            _next_free.push_back(-1);
        }
    }

    Node& operator[](int node) {
        assert(node != null_node && _nodes[node].has_value());
        return *_nodes[node];
    }

    const Node& operator[](int node) const {
        assert(node != null_node && _nodes[node].has_value());
        return *_nodes[node];
    }

    template <class... Args>
    int emplace(Args&&... args) {
        int result;
        if (_first_free == -1) {
            assert(_nodes.size() < std::size_t(std::numeric_limits<int>::max()));
            result = int(_nodes.size());
            _nodes.emplace_back(std::in_place, std::forward<Args>(args)...);
            _references.push_back(0);
            _next_free.push_back(-1);
        } else {
            result = _first_free;
            _first_free = _next_free[result];
            _nodes[result].emplace(std::forward<Args>(args)...);
            _references[result] = 0;
        }
        retain((*_nodes[result]).l);
        retain((*_nodes[result]).r);
        _unowned.push_back(result);
        ++_live_nodes;
        return result;
    }

    void retain(int node) {
        if (node != null_node) {
            assert(_nodes[node].has_value());
            ++_references[node];
        }
    }

    void release(int node) {
        if (node == null_node) return;
        assert(_nodes[node].has_value() && _references[node] > 0);
        if (--_references[node] == 0) release_zero(node);
    }

    bool unique(int node) const {
        return node == null_node || _references[node] == 1;
    }

    int clone(int node) {
        assert(node != null_node && _nodes[node].has_value());
        return emplace(*_nodes[node]);
    }

    // Returns node itself when it has one owner, otherwise an unowned clone.
    // A returned clone becomes owned when a root or parent edge retains it.
    int clone_if_shared(int node) {
        if (unique(node)) return node;
        return clone(node);
    }

    void replace(int& edge, int node) {
        if (edge == node) return;
        retain(node);
        int old = edge;
        edge = node;
        release(old);
    }

    void discard_unreferenced() {
        while (!_unowned.empty()) {
            int node = _unowned.back();
            _unowned.pop_back();
            if (_nodes[node].has_value() && _references[node] == 0) release_zero(node);
        }
    }

    void reserve(std::size_t) {}

    int next_index() const { return _first_free == -1 ? int(_nodes.size()) : _first_free; }

    std::size_t size() const { return _live_nodes; }
};

}  // namespace detail
}  // namespace ds
}  // namespace m1une


#line 12 "ds/dsu/persistent_dsu.hpp"

namespace m1une {
namespace ds {

struct PersistentDsu {
   private:
    struct Node {
        int val;
        int l, r;

        Node() : val(0), l(0), r(0) {}
        explicit Node(int value) : val(value), l(0), r(0) {}
        Node(int value, int left, int right) : val(value), l(left), r(right) {}
    };

    int _n;
    int _root;
    using Pool = detail::PersistentBinaryNodePool<Node, 0>;

    std::shared_ptr<Pool> _pool;

    explicit PersistentDsu(int n, int root, std::shared_ptr<Pool> pool)
        : _n(n), _root(root), _pool(std::move(pool)) {
        _pool->retain(_root);
    }

    int new_node(const Node& node) const {
        return _pool->emplace(node);
    }

    int new_node(Node&& node) const {
        return _pool->emplace(std::move(node));
    }

    int build(int l, int r) const {
        if (l == r) return 0;
        if (r - l == 1) return new_node(Node(-1));
        int m = (l + r) >> 1;
        int left = build(l, m);
        int right = build(m, r);
        return new_node(Node(0, left, right));
    }

    int set_node(int t, int l, int r, int p, int value, bool copy_on_write = false) const {
        if (copy_on_write) t = _pool->clone_if_shared(t);
        if (r - l == 1) {
            if (copy_on_write) {
                (*_pool)[t].val = value;
                return t;
            }
            return new_node(Node(value));
        }
        int m = (l + r) >> 1;
        int left = (*_pool)[t].l;
        int right = (*_pool)[t].r;
        if (p < m) {
            left = set_node(left, l, m, p, value, copy_on_write);
        } else {
            right = set_node(right, m, r, p, value, copy_on_write);
        }
        if (copy_on_write) {
            _pool->replace((*_pool)[t].l, left);
            _pool->replace((*_pool)[t].r, right);
            return t;
        }
        return new_node(Node(0, left, right));
    }

    PersistentDsu make_version(int root) const {
        PersistentDsu result(_n, root, _pool);
        _pool->discard_unreferenced();
        return result;
    }

    int get_node(int t, int l, int r, int p) const {
        while (r - l > 1) {
            int m = (l + r) >> 1;
            if (p < m) {
                t = (*_pool)[t].l;
                r = m;
            } else {
                t = (*_pool)[t].r;
                l = m;
            }
        }
        return (*_pool)[t].val;
    }

   public:
    PersistentDsu() : PersistentDsu(0) {}

    explicit PersistentDsu(int n) : _n(n), _root(0), _pool(std::make_shared<Pool>()) {
        assert(0 <= n);
        _pool->reserve(n * 2 + 1);
        if (_n > 0) _root = build(0, _n);
        _pool->retain(_root);
        _pool->discard_unreferenced();
    }

    PersistentDsu(const PersistentDsu& other) : _n(other._n), _root(other._root), _pool(other._pool) {
        if (_pool) _pool->retain(_root);
    }

    PersistentDsu(PersistentDsu&& other) noexcept
        : _n(other._n), _root(other._root), _pool(std::move(other._pool)) {
        other._n = 0;
        other._root = 0;
    }

    PersistentDsu& operator=(const PersistentDsu& other) {
        if (this == &other) return *this;
        if (other._pool) other._pool->retain(other._root);
        if (_pool) _pool->release(_root);
        _n = other._n;
        _root = other._root;
        _pool = other._pool;
        return *this;
    }

    PersistentDsu& operator=(PersistentDsu&& other) noexcept {
        if (this == &other) return *this;
        if (_pool) _pool->release(_root);
        _n = other._n;
        _root = other._root;
        _pool = std::move(other._pool);
        other._n = 0;
        other._root = 0;
        return *this;
    }

    ~PersistentDsu() {
        if (_pool) _pool->release(_root);
    }

    int size() const {
        return _n;
    }

    bool empty() const {
        return _n == 0;
    }

    void release() {
        if (_pool) _pool->release(_root);
        _n = 0;
        _root = 0;
        _pool = std::make_shared<Pool>();
    }

    std::size_t node_count() const { return _pool ? _pool->size() : 0; }

    int leader(int a) const {
        assert(0 <= a && a < _n);
        int x = a;
        int p = get(x);
        while (p >= 0) {
            x = p;
            p = get(x);
        }
        return x;
    }

    bool same(int a, int b) const {
        assert(0 <= a && a < _n);
        assert(0 <= b && b < _n);
        return leader(a) == leader(b);
    }

    int group_size(int a) const {
        assert(0 <= a && a < _n);
        return -get(leader(a));
    }

    int size(int a) const {
        return group_size(a);
    }

    int get(int p) const {
        assert(0 <= p && p < _n);
        return get_node(_root, 0, _n, p);
    }

    PersistentDsu merge(int a, int b) const {
        assert(0 <= a && a < _n);
        assert(0 <= b && b < _n);
        int x = leader(a), y = leader(b);
        if (x == y) return *this;
        int sx = -get(x), sy = -get(y);
        if (sx < sy) {
            std::swap(x, y);
            std::swap(sx, sy);
        }
        int root = set_node(_root, 0, _n, x, -(sx + sy));
        root = set_node(root, 0, _n, y, x);
        return make_version(root);
    }

    bool merge_inplace(int a, int b) {
        assert(0 <= a && a < _n);
        assert(0 <= b && b < _n);
        int x = leader(a), y = leader(b);
        if (x == y) return false;
        int sx = -get(x), sy = -get(y);
        if (sx < sy) {
            std::swap(x, y);
            std::swap(sx, sy);
        }
        int root = set_node(_root, 0, _n, x, -(sx + sy), true);
        _pool->replace(_root, root);
        root = set_node(_root, 0, _n, y, x, true);
        _pool->replace(_root, root);
        _pool->discard_unreferenced();
        return true;
    }

    std::vector<std::vector<int>> groups() const {
        std::vector<int> leader_buf(_n), group_size(_n);
        for (int i = 0; i < _n; i++) {
            leader_buf[i] = leader(i);
            group_size[leader_buf[i]]++;
        }
        std::vector<std::vector<int>> result(_n);
        for (int i = 0; i < _n; i++) {
            result[i].reserve(group_size[i]);
        }
        for (int i = 0; i < _n; i++) {
            result[leader_buf[i]].push_back(i);
        }
        result.erase(std::remove_if(result.begin(), result.end(), [&](const std::vector<int>& v) { return v.empty(); }),
                     result.end());
        return result;
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "ds/dsu/persistent_potentialized_dsu.hpp"



#line 6 "ds/dsu/persistent_potentialized_dsu.hpp"
#include <concepts>
#line 11 "ds/dsu/persistent_potentialized_dsu.hpp"

#line 1 "monoid/concept.hpp"



#line 5 "monoid/concept.hpp"

namespace m1une {
namespace monoid {

// Concept to check if a type satisfies the requirements of a Monoid.
// A Monoid must have a `value_type`, an identity element `id()`, and an associative binary operation `op()`.
template <typename M>
concept IsMonoid = requires(typename M::value_type a, typename M::value_type b) {
    // 1. Must define `value_type`
    typename M::value_type;

    // 2. Must have a static method `id()` returning `value_type`
    { M::id() } -> std::same_as<typename M::value_type>;

    // 3. Must have a static method `op(a, b)` returning `value_type`
    { M::op(a, b) } -> std::same_as<typename M::value_type>;
};

// Concept for groups. A type satisfying this concept must also obey the group
// laws; concepts can check the interface but not the algebraic properties.
template <typename M>
concept IsGroup = IsMonoid<M> && requires(typename M::value_type a) {
    { M::inv(a) } -> std::same_as<typename M::value_type>;
};

// Concept for commutative groups. Commutativity is a semantic requirement and
// cannot be checked by a C++ concept.
template <typename M>
concept IsCommutativeGroup = IsGroup<M>;

}  // namespace monoid
}  // namespace m1une


#line 14 "ds/dsu/persistent_potentialized_dsu.hpp"

namespace m1une {
namespace ds {

template <m1une::monoid::IsGroup Group>
    requires std::equality_comparable<typename Group::value_type>
struct PersistentPotentializedDsu {
    using T = typename Group::value_type;

    struct Value {
        int parent_or_size;
        T diff_to_parent;

        Value() : parent_or_size(0), diff_to_parent(Group::id()) {}
        Value(int parent_or_size_, const T& diff_to_parent_)
            : parent_or_size(parent_or_size_), diff_to_parent(diff_to_parent_) {}
        Value(int parent_or_size_, T&& diff_to_parent_)
            : parent_or_size(parent_or_size_), diff_to_parent(std::move(diff_to_parent_)) {}
    };

   private:
    struct Node {
        Value val;
        int l, r;

        Node() : val(), l(0), r(0) {}
        explicit Node(const Value& value) : val(value), l(0), r(0) {}
        explicit Node(Value&& value) : val(std::move(value)), l(0), r(0) {}
        Node(const Value& value, int left, int right) : val(value), l(left), r(right) {}
        Node(Value&& value, int left, int right) : val(std::move(value)), l(left), r(right) {}
    };

    int _n;
    int _root;
    using Pool = detail::PersistentBinaryNodePool<Node, 0>;

    std::shared_ptr<Pool> _pool;

    explicit PersistentPotentializedDsu(int n, int root, std::shared_ptr<Pool> pool)
        : _n(n), _root(root), _pool(std::move(pool)) {
        _pool->retain(_root);
    }

    int new_node(const Node& node) const {
        return _pool->emplace(node);
    }

    int new_node(Node&& node) const {
        return _pool->emplace(std::move(node));
    }

    int build(int l, int r) const {
        if (l == r) return 0;
        if (r - l == 1) return new_node(Node(Value(-1, Group::id())));
        int m = (l + r) >> 1;
        int left = build(l, m);
        int right = build(m, r);
        return new_node(Node(Value(), left, right));
    }

    int set_node(int t, int l, int r, int p, Value value, bool copy_on_write = false) const {
        if (copy_on_write) t = _pool->clone_if_shared(t);
        if (r - l == 1) {
            if (copy_on_write) {
                (*_pool)[t].val = std::move(value);
                return t;
            }
            return new_node(Node(std::move(value)));
        }
        int m = (l + r) >> 1;
        int left = (*_pool)[t].l;
        int right = (*_pool)[t].r;
        if (p < m) {
            left = set_node(left, l, m, p, std::move(value), copy_on_write);
        } else {
            right = set_node(right, m, r, p, std::move(value), copy_on_write);
        }
        if (copy_on_write) {
            _pool->replace((*_pool)[t].l, left);
            _pool->replace((*_pool)[t].r, right);
            return t;
        }
        return new_node(Node(Value(), left, right));
    }

    Value get_value(int t, int l, int r, int p) const {
        while (r - l > 1) {
            int m = (l + r) >> 1;
            if (p < m) {
                t = (*_pool)[t].l;
                r = m;
            } else {
                t = (*_pool)[t].r;
                l = m;
            }
        }
        return (*_pool)[t].val;
    }

    std::pair<int, T> leader_and_potential(int a) const {
        T res = Group::id();
        while (true) {
            Value cur = get(a);
            if (cur.parent_or_size < 0) return {a, res};
            res = Group::op(cur.diff_to_parent, res);
            a = cur.parent_or_size;
        }
    }

    PersistentPotentializedDsu make_version(int root) const {
        PersistentPotentializedDsu result(_n, root, _pool);
        _pool->discard_unreferenced();
        return result;
    }

   public:
    PersistentPotentializedDsu() : PersistentPotentializedDsu(0) {}

    explicit PersistentPotentializedDsu(int n) : _n(n), _root(0), _pool(std::make_shared<Pool>()) {
        assert(0 <= n);
        _pool->reserve(n * 4 + 1);
        if (_n > 0) _root = build(0, _n);
        _pool->retain(_root);
        _pool->discard_unreferenced();
    }

    PersistentPotentializedDsu(const PersistentPotentializedDsu& other)
        : _n(other._n), _root(other._root), _pool(other._pool) {
        if (_pool) _pool->retain(_root);
    }

    PersistentPotentializedDsu(PersistentPotentializedDsu&& other) noexcept
        : _n(other._n), _root(other._root), _pool(std::move(other._pool)) {
        other._n = 0;
        other._root = 0;
    }

    PersistentPotentializedDsu& operator=(const PersistentPotentializedDsu& other) {
        if (this == &other) return *this;
        if (other._pool) other._pool->retain(other._root);
        if (_pool) _pool->release(_root);
        _n = other._n;
        _root = other._root;
        _pool = other._pool;
        return *this;
    }

    PersistentPotentializedDsu& operator=(PersistentPotentializedDsu&& other) noexcept {
        if (this == &other) return *this;
        if (_pool) _pool->release(_root);
        _n = other._n;
        _root = other._root;
        _pool = std::move(other._pool);
        other._n = 0;
        other._root = 0;
        return *this;
    }

    ~PersistentPotentializedDsu() {
        if (_pool) _pool->release(_root);
    }

    int size() const {
        return _n;
    }

    bool empty() const {
        return _n == 0;
    }

    void release() {
        if (_pool) _pool->release(_root);
        _n = 0;
        _root = 0;
        _pool = std::make_shared<Pool>();
    }

    std::size_t node_count() const { return _pool ? _pool->size() : 0; }

    int leader(int a) const {
        assert(0 <= a && a < _n);
        return leader_and_potential(a).first;
    }

    bool same(int a, int b) const {
        assert(0 <= a && a < _n);
        assert(0 <= b && b < _n);
        return leader(a) == leader(b);
    }

    int group_size(int a) const {
        assert(0 <= a && a < _n);
        return -get(leader(a)).parent_or_size;
    }

    int size(int a) const {
        return group_size(a);
    }

    T potential(int a) const {
        assert(0 <= a && a < _n);
        return leader_and_potential(a).second;
    }

    T diff(int a, int b) const {
        assert(same(a, b));
        return Group::op(Group::inv(potential(a)), potential(b));
    }

    Value get(int p) const {
        assert(0 <= p && p < _n);
        return get_value(_root, 0, _n, p);
    }

    int parent_or_size(int p) const {
        return get(p).parent_or_size;
    }

    std::pair<PersistentPotentializedDsu, bool> merge(int a, int b, const T& w) const {
        assert(0 <= a && a < _n);
        assert(0 <= b && b < _n);
        auto [x, pa] = leader_and_potential(a);
        auto [y, pb] = leader_and_potential(b);
        if (x == y) return {*this, Group::op(Group::inv(pa), pb) == w};

        int sx = -get(x).parent_or_size;
        int sy = -get(y).parent_or_size;
        T y_from_x = Group::op(Group::op(pa, w), Group::inv(pb));
        if (sx < sy) {
            std::swap(x, y);
            std::swap(sx, sy);
            y_from_x = Group::inv(y_from_x);
        }
        int root = set_node(_root, 0, _n, x, Value(-(sx + sy), Group::id()));
        root = set_node(root, 0, _n, y, Value(x, std::move(y_from_x)));
        return {make_version(root), true};
    }

    bool merge_inplace(int a, int b, const T& w) {
        assert(0 <= a && a < _n);
        assert(0 <= b && b < _n);
        auto [x, pa] = leader_and_potential(a);
        auto [y, pb] = leader_and_potential(b);
        if (x == y) return Group::op(Group::inv(pa), pb) == w;

        int sx = -get(x).parent_or_size;
        int sy = -get(y).parent_or_size;
        T y_from_x = Group::op(Group::op(pa, w), Group::inv(pb));
        if (sx < sy) {
            std::swap(x, y);
            std::swap(sx, sy);
            y_from_x = Group::inv(y_from_x);
        }
        int root = set_node(_root, 0, _n, x, Value(-(sx + sy), Group::id()), true);
        _pool->replace(_root, root);
        root = set_node(_root, 0, _n, y, Value(x, std::move(y_from_x)), true);
        _pool->replace(_root, root);
        _pool->discard_unreferenced();
        return true;
    }

    std::vector<std::vector<int>> groups() const {
        std::vector<int> leader_buf(_n), group_size(_n);
        for (int i = 0; i < _n; i++) {
            leader_buf[i] = leader(i);
            group_size[leader_buf[i]]++;
        }
        std::vector<std::vector<int>> result(_n);
        for (int i = 0; i < _n; i++) {
            result[i].reserve(group_size[i]);
        }
        for (int i = 0; i < _n; i++) {
            result[leader_buf[i]].push_back(i);
        }
        result.erase(std::remove_if(result.begin(), result.end(), [&](const std::vector<int>& v) { return v.empty(); }),
                     result.end());
        return result;
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "ds/bst/persistent_ordered_multiset.hpp"



#line 6 "ds/bst/persistent_ordered_multiset.hpp"
#include <functional>
#include <initializer_list>
#line 10 "ds/bst/persistent_ordered_multiset.hpp"

#line 12 "ds/bst/persistent_ordered_multiset.hpp"

namespace m1une {
namespace ds {

template <typename T, typename Compare>
struct PersistentOrderedSet;

template <typename T, typename Compare = std::less<T>>
struct PersistentOrderedMultiset {
   private:
    friend struct PersistentOrderedSet<T, Compare>;
    struct Node {
        T key;
        int count;
        int size;
        int distinct_size;
        int rank_color;
        int l;
        int r;
        int min_leaf;
        int max_leaf;

        Node(T value, int multiplicity, int maximum)
            : key(std::move(value)),
              count(multiplicity),
              size(multiplicity),
              distinct_size(1),
              rank_color(1),
              l(-1),
              r(-1),
              min_leaf(maximum),
              max_leaf(maximum) {}

        Node(T separator, int subtree_size, int left_size, int unique_count, int node_rank,
             int left, int right, int minimum, int maximum, bool is_black)
            : key(std::move(separator)),
              count(left_size),
              size(subtree_size),
              distinct_size(unique_count),
              rank_color(node_rank * 2 + int(is_black)),
              l(left),
              r(right),
              min_leaf(minimum),
              max_leaf(maximum) {}
    };

    using Pool = detail::PersistentBinaryNodePool<Node>;

    inline static Pool pool;

    int root;
    Compare comp;

    static int subtree_size(int t) { return t == -1 ? 0 : pool[t].size; }
    static int subtree_distinct_size(int t) { return t == -1 ? 0 : pool[t].distinct_size; }
    static int node_rank(int t) { return pool[t].rank_color >> 1; }
    static bool is_black(int t) { return (pool[t].rank_color & 1) != 0; }
    static bool is_leaf(int t) { return pool[t].l == -1; }

    bool equal(const T& a, const T& b) const {
        return !comp(a, b) && !comp(b, a);
    }

    static int make_leaf(T key, int count) {
        const int id = pool.next_index();
        return pool.emplace(std::move(key), count, id);
    }

    static int make_node(int l, int r, bool black) {
        assert(l != -1 && r != -1);
        const int rank = node_rank(l) + int(is_black(l));
        assert(rank == node_rank(r) + int(is_black(r)));
        return pool.emplace(pool[pool[l].max_leaf].key,
                            subtree_size(l) + subtree_size(r),
                            subtree_size(l),
                            subtree_distinct_size(l) + subtree_distinct_size(r),
                            rank, l, r, pool[l].min_leaf, pool[r].max_leaf,
                            black);
    }

    static int as_root(int t) {
        if (t == -1 || is_black(t)) return t;
        return make_node(pool[t].l, pool[t].r, true);
    }

    static int merge_sub(int a, int b) {
        assert(a != -1 && b != -1);
        if (node_rank(a) < node_rank(b)) {
            const Node& right = pool[b];
            int c = merge_sub(a, right.l);
            if (is_black(b) && !is_black(c) && !is_black(pool[c].l)) {
                const Node& middle = pool[c];
                if (is_black(right.r)) {
                    return make_node(middle.l,
                                     make_node(middle.r, right.r, false),
                                     true);
                }
                const Node& far = pool[right.r];
                return make_node(make_node(middle.l, middle.r, true),
                                 make_node(far.l, far.r, true),
                                 false);
            }
            return make_node(c, right.r, is_black(b));
        }
        if (node_rank(a) > node_rank(b)) {
            const Node& left = pool[a];
            int c = merge_sub(left.r, b);
            if (is_black(a) && !is_black(c) && !is_black(pool[c].r)) {
                const Node& middle = pool[c];
                if (is_black(left.l)) {
                    return make_node(make_node(left.l, middle.l, false),
                                     middle.r, true);
                }
                const Node& far = pool[left.l];
                return make_node(make_node(far.l, far.r, true),
                                 make_node(middle.l, middle.r, true),
                                 false);
            }
            return make_node(left.l, c, is_black(a));
        }
        return make_node(a, b, false);
    }

    static int merge_nodes(int a, int b) {
        if (a == -1 || b == -1) return a == -1 ? b : a;
        return as_root(merge_sub(as_root(a), as_root(b)));
    }

    std::pair<int, int> split_nodes(int t, const T& key) const {
        if (t == -1) return {-1, -1};
        const Node& node = pool[t];
        if (is_leaf(t)) return comp(node.key, key) ? std::pair{t, -1} : std::pair{-1, t};
        if (comp(node.key, key)) {
            auto [l, r] = split_nodes(node.r, key);
            return {merge_nodes(as_root(node.l), l), r};
        }
        auto [l, r] = split_nodes(node.l, key);
        return {l, merge_nodes(r, as_root(node.r))};
    }

    int change_count_impl(int t, const T& key, int delta, int& old_count) const {
        if (t == -1) return -1;
        const Node& node = pool[t];
        if (is_leaf(t)) {
            if (!equal(node.key, key)) return t;
            old_count = node.count;
            return node.count + delta == 0 ? t : make_leaf(node.key, node.count + delta);
        }
        int child;
        if (!comp(node.key, key)) {
            child = change_count_impl(node.l, key, delta, old_count);
            if (old_count == 0 || old_count + delta == 0) return t;
            return make_node(child, node.r, is_black(t));
        }
        child = change_count_impl(node.r, key, delta, old_count);
        if (old_count == 0 || old_count + delta == 0) return t;
        return make_node(node.l, child, is_black(t));
    }

    int change_count_inplace(int t, const T& key, int delta) const {
        t = pool.clone_if_shared(t);
        if (is_leaf(t)) {
            assert(equal(pool[t].key, key));
            assert(pool[t].count + delta > 0);
            pool[t].count += delta;
            pool[t].size += delta;
            return t;
        }
        if (!comp(pool[t].key, key)) {
            int child = change_count_inplace(pool[t].l, key, delta);
            pool.replace(pool[t].l, child);
        } else {
            int child = change_count_inplace(pool[t].r, key, delta);
            pool.replace(pool[t].r, child);
        }
        Node& node = pool[t];
        node.count = subtree_size(node.l);
        node.size = node.count + subtree_size(node.r);
        return t;
    }

    int count_impl(int t, const T& key) const {
        if (t == -1) return 0;
        while (!is_leaf(t)) {
            t = !comp(pool[t].key, key) ? pool[t].l : pool[t].r;
        }
        return equal(pool[t].key, key) ? pool[t].count : 0;
    }

    const T* kth_impl(int t, int k) const {
        while (!is_leaf(t)) {
            const int left_size = pool[t].count;
            if (k < left_size) {
                t = pool[t].l;
            } else {
                k -= left_size;
                t = pool[t].r;
            }
        }
        return &pool[t].key;
    }

    int order_of_key_impl(int t, const T& key, bool upper) const {
        int result = 0;
        while (t != -1 && !is_leaf(t)) {
            const Node& node = pool[t];
            const T& separator = node.key;
            const bool take_left = upper ? !comp(key, separator) : comp(separator, key);
            if (take_left) {
                result += node.count;
                t = node.r;
            } else {
                t = node.l;
            }
        }
        if (t != -1) {
            const bool take_leaf = upper ? !comp(key, pool[t].key) : comp(pool[t].key, key);
            if (take_leaf) result += pool[t].count;
        }
        return result;
    }

    const T* lower_bound_impl(int t, const T& key, bool strict) const {
        const T* result = nullptr;
        while (t != -1) {
            const Node& node = pool[t];
            if (is_leaf(t)) {
                const bool candidate = strict ? comp(key, node.key) : !comp(node.key, key);
                return candidate ? &node.key : result;
            }
            const T& separator = node.key;
            const bool go_left = strict ? comp(key, separator) : !comp(separator, key);
            if (go_left) {
                result = &pool[pool[node.r].min_leaf].key;
                t = node.l;
            } else {
                t = node.r;
            }
        }
        return result;
    }

    const T* max_less_impl(int t, const T& key, bool strict) const {
        const T* result = nullptr;
        while (t != -1) {
            const Node& node = pool[t];
            if (is_leaf(t)) {
                const bool candidate = strict ? comp(node.key, key) : !comp(key, node.key);
                return candidate ? &node.key : result;
            }
            const T& separator = node.key;
            const bool take_left = strict ? comp(separator, key) : !comp(key, separator);
            if (take_left) {
                result = &separator;
                t = node.r;
            } else {
                t = node.l;
            }
        }
        return result;
    }

    static void dump_impl(int t, std::vector<T>& result) {
        if (t == -1) return;
        const Node& node = pool[t];
        if (is_leaf(t)) {
            for (int i = 0; i < node.count; ++i) result.push_back(node.key);
            return;
        }
        dump_impl(node.l, result);
        dump_impl(node.r, result);
    }

    static std::pair<int, int> pop_min(int t) {
        assert(t != -1);
        const Node& node = pool[t];
        if (is_leaf(t)) return {t, -1};
        auto [minimum, rest] = pop_min(node.l);
        return {minimum, merge_nodes(rest, as_root(node.r))};
    }

    PersistentOrderedMultiset(int node, Compare compare) : root(node), comp(std::move(compare)) {
        pool.retain(root);
    }

    PersistentOrderedMultiset make_version(int node) const {
        PersistentOrderedMultiset result(node, comp);
        pool.discard_unreferenced();
        return result;
    }

   public:
    explicit PersistentOrderedMultiset(Compare compare) : root(-1), comp(std::move(compare)) {}
    PersistentOrderedMultiset() : PersistentOrderedMultiset(Compare()) {}

    PersistentOrderedMultiset(std::initializer_list<T> init, Compare compare = Compare())
        : PersistentOrderedMultiset(std::move(compare)) {
        for (const T& x : init) *this = insert(x);
    }

    template <typename Iterator>
    PersistentOrderedMultiset(Iterator first, Iterator last, Compare compare = Compare())
        : PersistentOrderedMultiset(std::move(compare)) {
        while (first != last) *this = insert(*first++);
    }

    PersistentOrderedMultiset(const PersistentOrderedMultiset& other)
        : root(other.root), comp(other.comp) {
        pool.retain(root);
    }

    PersistentOrderedMultiset(PersistentOrderedMultiset&& other)
        : root(other.root), comp(std::move(other.comp)) {
        other.root = -1;
    }

    PersistentOrderedMultiset& operator=(const PersistentOrderedMultiset& other) {
        if (this == &other) return *this;
        pool.retain(other.root);
        pool.release(root);
        root = other.root;
        comp = other.comp;
        return *this;
    }

    PersistentOrderedMultiset& operator=(PersistentOrderedMultiset&& other) {
        if (this == &other) return *this;
        pool.release(root);
        root = other.root;
        comp = std::move(other.comp);
        other.root = -1;
        return *this;
    }

    ~PersistentOrderedMultiset() { pool.release(root); }

    int size() const { return subtree_size(root); }
    int unique_size() const { return subtree_distinct_size(root); }
    bool empty() const { return root == -1; }
    void release() { pool.release(std::exchange(root, -1)); }
    std::size_t node_count() const { return pool.size(); }
    PersistentOrderedMultiset clear() const { return make_version(-1); }

    PersistentOrderedMultiset insert(T key, int multiplicity = 1) const {
        assert(multiplicity > 0);
        int old_count = 0;
        const int changed_root = change_count_impl(root, key, multiplicity, old_count);
        if (old_count != 0) {
            return make_version(changed_root);
        }
        auto [l, r] = split_nodes(root, key);
        return make_version(merge_nodes(merge_nodes(l, make_leaf(std::move(key), multiplicity)), r));
    }

    void insert_inplace(T key, int multiplicity = 1) {
        assert(multiplicity > 0);
        if (!contains(key)) {
            *this = insert(std::move(key), multiplicity);
            return;
        }
        int next_root = change_count_inplace(root, key, multiplicity);
        pool.replace(root, next_root);
        pool.discard_unreferenced();
    }

   private:
    PersistentOrderedMultiset insert_unique(T key) const {
        if (contains(key)) return *this;
        auto [l, r] = split_nodes(root, key);
        return make_version(merge_nodes(merge_nodes(l, make_leaf(std::move(key), 1)), r));
    }

   public:
    PersistentOrderedMultiset erase_one(const T& key) const {
        int old_count = 0;
        const int changed_root = change_count_impl(root, key, -1, old_count);
        if (old_count == 0) return *this;
        if (old_count > 1) return make_version(changed_root);
        auto [l, r] = split_nodes(root, key);
        auto [discarded, rest] = pop_min(r);
        assert(equal(pool[discarded].key, key));
        return make_version(merge_nodes(l, rest));
    }

    PersistentOrderedMultiset erase(const T& key) const { return erase_one(key); }

    bool erase_one_inplace(const T& key) {
        int old_count = count(key);
        if (old_count == 0) return false;
        if (old_count == 1) {
            *this = erase_one(key);
            return true;
        }
        int next_root = change_count_inplace(root, key, -1);
        pool.replace(root, next_root);
        pool.discard_unreferenced();
        return true;
    }

    bool erase_inplace(const T& key) { return erase_one_inplace(key); }

    PersistentOrderedMultiset erase_all(const T& key) const {
        const int old_count = count(key);
        if (old_count == 0) return *this;
        auto [l, r] = split_nodes(root, key);
        auto [discarded, rest] = pop_min(r);
        assert(equal(pool[discarded].key, key));
        return make_version(merge_nodes(l, rest));
    }

    bool erase_all_inplace(const T& key) {
        if (!contains(key)) return false;
        *this = erase_all(key);
        return true;
    }

    bool contains(const T& key) const { return count(key) > 0; }
    int count(const T& key) const { return count_impl(root, key); }

    const T* find_by_order(int k) const {
        assert(0 <= k && k < size());
        return kth_impl(root, k);
    }

    T kth(int k) const { return *find_by_order(k); }
    int order_of_key(const T& key) const { return order_of_key_impl(root, key, false); }
    int count_less(const T& key) const { return order_of_key(key); }
    int count_less_equal(const T& key) const { return order_of_key_impl(root, key, true); }
    int count_greater(const T& key) const { return size() - count_less_equal(key); }
    int count_greater_equal(const T& key) const { return size() - count_less(key); }
    const T* lower_bound(const T& key) const { return lower_bound_impl(root, key, false); }
    const T* upper_bound(const T& key) const { return lower_bound_impl(root, key, true); }
    const T* min_ge(const T& key) const { return lower_bound(key); }
    const T* min_gt(const T& key) const { return upper_bound(key); }
    const T* max_le(const T& key) const { return max_less_impl(root, key, false); }
    const T* max_lt(const T& key) const { return max_less_impl(root, key, true); }
    const T* min() const { return empty() ? nullptr : &pool[pool[root].min_leaf].key; }
    const T* max() const { return empty() ? nullptr : &pool[pool[root].max_leaf].key; }

    std::pair<PersistentOrderedMultiset, PersistentOrderedMultiset> split(const T& key) const {
        auto [l, r] = split_nodes(root, key);
        PersistentOrderedMultiset left(l, comp);
        PersistentOrderedMultiset right(r, comp);
        pool.discard_unreferenced();
        return {std::move(left), std::move(right)};
    }

    PersistentOrderedMultiset merge(const PersistentOrderedMultiset& other) const {
        assert(empty() || other.empty() || comp(*max(), *other.min()));
        return make_version(merge_nodes(root, other.root));
    }

    std::vector<T> to_vector() const {
        std::vector<T> result;
        result.reserve(size());
        dump_impl(root, result);
        return result;
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "ds/dynamic_array/persistent_dynamic_array.hpp"



#line 5 "ds/dynamic_array/persistent_dynamic_array.hpp"
#include <chrono>
#line 7 "ds/dynamic_array/persistent_dynamic_array.hpp"
#include <cstdint>
#line 13 "ds/dynamic_array/persistent_dynamic_array.hpp"

#line 15 "ds/dynamic_array/persistent_dynamic_array.hpp"

namespace m1une {
namespace ds {

template <typename T>
struct PersistentDynamicArray {
   private:
    struct Node {
        T val;
        int priority;
        int count;
        int l, r;
        bool rev;

        Node(T value, int node_priority, int node_count, int left, int right, bool reversed)
            : val(std::move(value)),
              priority(node_priority),
              count(node_count),
              l(left),
              r(right),
              rev(reversed) {}
    };

    struct BuildNode {
        T val;
        int priority;
        int l, r;

        BuildNode(T value, int node_priority) : val(std::move(value)), priority(node_priority), l(-1), r(-1) {}
    };

    int root;
    std::uint32_t rng_state;
    using Pool = detail::PersistentBinaryNodePool<Node>;

    std::shared_ptr<Pool> pool;

    int subtree_size(int t) const {
        return t == -1 ? 0 : (*pool)[t].count;
    }

    static std::uint32_t next_state(std::uint32_t state) {
        state ^= state << 13;
        state ^= state >> 17;
        state ^= state << 5;
        return state == 0 ? 1 : state;
    }

    static int next_priority(std::uint32_t& state) {
        state = next_state(state);
        return int(state);
    }

    int make_node(T val, int priority, bool rev, int l, int r) const {
        int count = 1 + subtree_size(l) + subtree_size(r);
        return pool->emplace(std::move(val), priority, count, l, r, rev);
    }

    int reversed_node(int t) const {
        if (t == -1) return -1;
        const Node& node = (*pool)[t];
        return make_node(node.val, node.priority, !node.rev, node.l, node.r);
    }

    int push(int t) const {
        if (t == -1 || !(*pool)[t].rev) return t;
        Node node = (*pool)[t];
        int l = reversed_node(node.r);
        int r = reversed_node(node.l);
        return make_node(std::move(node.val), node.priority, false, l, r);
    }

    int merge(int l, int r) const {
        if (l == -1 || r == -1) return l == -1 ? r : l;
        if ((*pool)[l].priority > (*pool)[r].priority) {
            Node node = (*pool)[push(l)];
            int right = merge(node.r, r);
            return make_node(std::move(node.val), node.priority, false, node.l, right);
        }
        Node node = (*pool)[push(r)];
        int left = merge(l, node.l);
        return make_node(std::move(node.val), node.priority, false, left, node.r);
    }

    std::pair<int, int> split_node(int t, int pos) const {
        if (t == -1) return {-1, -1};
        Node node = (*pool)[push(t)];
        int left_count = subtree_size(node.l);
        if (pos <= left_count) {
            auto [a, b] = split_node(node.l, pos);
            return {a, make_node(std::move(node.val), node.priority, false, b, node.r)};
        }
        auto [a, b] = split_node(node.r, pos - left_count - 1);
        return {make_node(std::move(node.val), node.priority, false, node.l, a), b};
    }

    int set_node(int t, int pos, T val) const {
        Node node = (*pool)[push(t)];
        int left_count = subtree_size(node.l);
        if (pos < left_count) {
            int l = set_node(node.l, pos, std::move(val));
            return make_node(std::move(node.val), node.priority, false, l, node.r);
        }
        if (pos == left_count) {
            return make_node(std::move(val), node.priority, false, node.l, node.r);
        }
        int r = set_node(node.r, pos - left_count - 1, std::move(val));
        return make_node(std::move(node.val), node.priority, false, node.l, r);
    }

    int set_node_inplace(int t, int pos, T val, bool inherited_reversed = false) const {
        t = pool->clone_if_shared(t);
        const bool reversed = inherited_reversed ^ (*pool)[t].rev;
        const int logical_left = reversed ? (*pool)[t].r : (*pool)[t].l;
        const int left_count = subtree_size(logical_left);
        if (pos < left_count) {
            int child = set_node_inplace(logical_left, pos, std::move(val), reversed);
            if (reversed) {
                pool->replace((*pool)[t].r, child);
            } else {
                pool->replace((*pool)[t].l, child);
            }
        } else if (pos == left_count) {
            (*pool)[t].val = std::move(val);
        } else {
            const int logical_right = reversed ? (*pool)[t].l : (*pool)[t].r;
            int child = set_node_inplace(logical_right, pos - left_count - 1, std::move(val), reversed);
            if (reversed) {
                pool->replace((*pool)[t].l, child);
            } else {
                pool->replace((*pool)[t].r, child);
            }
        }
        return t;
    }

    int find_node(int t, int pos) const {
        bool reversed = false;
        while (t != -1) {
            const Node& node = (*pool)[t];
            bool cur_reversed = reversed ^ node.rev;
            int l = cur_reversed ? node.r : node.l;
            int r = cur_reversed ? node.l : node.r;
            int left_count = subtree_size(l);
            if (pos < left_count) {
                t = l;
                reversed = cur_reversed;
            } else if (pos == left_count) {
                return t;
            } else {
                pos -= left_count + 1;
                t = r;
                reversed = cur_reversed;
            }
        }
        return -1;
    }

    void dump_dfs(int t, std::vector<T>& res, bool reversed = false) const {
        if (t == -1) return;
        const Node& node = (*pool)[t];
        bool cur_reversed = reversed ^ node.rev;
        int l = cur_reversed ? node.r : node.l;
        int r = cur_reversed ? node.l : node.r;
        dump_dfs(l, res, cur_reversed);
        res.push_back(node.val);
        dump_dfs(r, res, cur_reversed);
    }

    void dump_range_dfs(int t, int ql, int qr, int offset, std::vector<T>& res, bool reversed = false) const {
        if (t == -1 || qr <= offset || offset + (*pool)[t].count <= ql) return;
        const Node& node = (*pool)[t];
        bool cur_reversed = reversed ^ node.rev;
        int l = cur_reversed ? node.r : node.l;
        int r = cur_reversed ? node.l : node.r;
        int left_count = subtree_size(l);
        int node_pos = offset + left_count;
        dump_range_dfs(l, ql, qr, offset, res, cur_reversed);
        if (ql <= node_pos && node_pos < qr) res.push_back(node.val);
        dump_range_dfs(r, ql, qr, node_pos + 1, res, cur_reversed);
    }

    int build_from_nodes(std::vector<BuildNode>& nodes, int t) const {
        if (t == -1) return -1;
        int l = build_from_nodes(nodes, nodes[t].l);
        int r = build_from_nodes(nodes, nodes[t].r);
        return make_node(std::move(nodes[t].val), nodes[t].priority, false, l, r);
    }

    int build_cartesian(std::vector<BuildNode>& nodes) const {
        if (nodes.empty()) return -1;
        std::vector<int> stack;
        stack.reserve(nodes.size());
        for (int i = 0; i < int(nodes.size()); i++) {
            int left_child = -1;
            while (!stack.empty() && nodes[stack.back()].priority < nodes[i].priority) {
                left_child = stack.back();
                stack.pop_back();
            }
            nodes[i].l = left_child;
            if (!stack.empty()) nodes[stack.back()].r = i;
            stack.push_back(i);
        }
        return build_from_nodes(nodes, stack.front());
    }

    int build_from_vector(const std::vector<T>& v, std::uint32_t& state) const {
        std::vector<BuildNode> nodes;
        nodes.reserve(v.size());
        for (const T& x : v) nodes.emplace_back(x, next_priority(state));
        return build_cartesian(nodes);
    }

    int build_from_vector(std::vector<T>&& v, std::uint32_t& state) const {
        std::vector<BuildNode> nodes;
        nodes.reserve(v.size());
        for (T& x : v) nodes.emplace_back(std::move(x), next_priority(state));
        return build_cartesian(nodes);
    }

    int import_node(const PersistentDynamicArray& other, int t) const {
        if (t == -1) return -1;
        if (pool == other.pool) return t;
        const Node& node = (*other.pool)[t];
        int l = import_node(other, node.l);
        int r = import_node(other, node.r);
        return make_node(node.val, node.priority, node.rev, l, r);
    }

    explicit PersistentDynamicArray(int node, std::uint32_t state, std::shared_ptr<Pool> node_pool)
        : root(node), rng_state(state), pool(std::move(node_pool)) {
        pool->retain(root);
    }

    PersistentDynamicArray make_version(int node, std::uint32_t state) const {
        PersistentDynamicArray result(node, state, pool);
        pool->discard_unreferenced();
        return result;
    }

   public:
    PersistentDynamicArray()
        : root(-1),
          rng_state(std::uint32_t(std::chrono::steady_clock::now().time_since_epoch().count())),
          pool(std::make_shared<Pool>()) {
        if (rng_state == 0) rng_state = 1;
    }

    explicit PersistentDynamicArray(int n) : PersistentDynamicArray(n, T()) {}

    PersistentDynamicArray(int n, const T& value) : PersistentDynamicArray() {
        assert(0 <= n);
        std::vector<T> v(n, value);
        root = build_from_vector(std::move(v), rng_state);
        pool->retain(root);
        pool->discard_unreferenced();
    }

    explicit PersistentDynamicArray(const std::vector<T>& v) : PersistentDynamicArray() {
        root = build_from_vector(v, rng_state);
        pool->retain(root);
        pool->discard_unreferenced();
    }

    explicit PersistentDynamicArray(std::vector<T>&& v) : PersistentDynamicArray() {
        root = build_from_vector(std::move(v), rng_state);
        pool->retain(root);
        pool->discard_unreferenced();
    }

    PersistentDynamicArray(std::initializer_list<T> init) : PersistentDynamicArray(std::vector<T>(init)) {}

    PersistentDynamicArray(const PersistentDynamicArray& other)
        : root(other.root), rng_state(other.rng_state), pool(other.pool) {
        if (pool) pool->retain(root);
    }

    PersistentDynamicArray(PersistentDynamicArray&& other) noexcept
        : root(other.root), rng_state(other.rng_state), pool(std::move(other.pool)) {
        other.root = -1;
    }

    PersistentDynamicArray& operator=(const PersistentDynamicArray& other) {
        if (this == &other) return *this;
        if (other.pool) other.pool->retain(other.root);
        if (pool) pool->release(root);
        root = other.root;
        rng_state = other.rng_state;
        pool = other.pool;
        return *this;
    }

    PersistentDynamicArray& operator=(PersistentDynamicArray&& other) noexcept {
        if (this == &other) return *this;
        if (pool) pool->release(root);
        root = other.root;
        rng_state = other.rng_state;
        pool = std::move(other.pool);
        other.root = -1;
        return *this;
    }

    ~PersistentDynamicArray() {
        if (pool) pool->release(root);
    }

    int size() const {
        return subtree_size(root);
    }

    bool empty() const {
        return size() == 0;
    }

    void release() {
        if (pool) pool->release(root);
        root = -1;
        pool = std::make_shared<Pool>();
    }

    std::size_t node_count() const { return pool ? pool->size() : 0; }

    PersistentDynamicArray clear() const {
        return make_version(-1, rng_state);
    }

    PersistentDynamicArray insert(int pos, T val) const {
        assert(0 <= pos && pos <= size());
        std::uint32_t next = next_state(rng_state);
        int node = make_node(std::move(val), int(next), false, -1, -1);
        auto [l, r] = split_node(root, pos);
        return make_version(merge(merge(l, node), r), next);
    }

    PersistentDynamicArray insert(int pos, const std::vector<T>& v) const {
        assert(0 <= pos && pos <= size());
        if (v.empty()) return *this;
        std::uint32_t next = rng_state;
        int mid = build_from_vector(v, next);
        auto [l, r] = split_node(root, pos);
        return make_version(merge(merge(l, mid), r), next);
    }

    PersistentDynamicArray insert(int pos, std::vector<T>&& v) const {
        assert(0 <= pos && pos <= size());
        if (v.empty()) return *this;
        std::uint32_t next = rng_state;
        int mid = build_from_vector(std::move(v), next);
        auto [l, r] = split_node(root, pos);
        return make_version(merge(merge(l, mid), r), next);
    }

    PersistentDynamicArray insert(int pos, std::initializer_list<T> init) const {
        return insert(pos, std::vector<T>(init));
    }

    PersistentDynamicArray insert(int pos, const PersistentDynamicArray& other) const {
        assert(0 <= pos && pos <= size());
        if (other.empty()) return *this;
        int mid = import_node(other, other.root);
        auto [l, r] = split_node(root, pos);
        return make_version(merge(merge(l, mid), r), rng_state);
    }

    PersistentDynamicArray push_back(T val) const {
        return insert(size(), std::move(val));
    }

    PersistentDynamicArray push_front(T val) const {
        return insert(0, std::move(val));
    }

    PersistentDynamicArray append(const std::vector<T>& v) const {
        return insert(size(), v);
    }

    PersistentDynamicArray append(std::vector<T>&& v) const {
        return insert(size(), std::move(v));
    }

    PersistentDynamicArray append(const PersistentDynamicArray& other) const {
        return insert(size(), other);
    }

    PersistentDynamicArray erase(int pos) const {
        assert(0 <= pos && pos < size());
        auto [a, b] = split_node(root, pos);
        auto [mid, c] = split_node(b, 1);
        (void)mid;
        return make_version(merge(a, c), rng_state);
    }

    PersistentDynamicArray erase(int l, int r) const {
        assert(0 <= l && l <= r && r <= size());
        if (l == r) return *this;
        auto [a, b] = split_node(root, l);
        auto [mid, c] = split_node(b, r - l);
        (void)mid;
        return make_version(merge(a, c), rng_state);
    }

    PersistentDynamicArray pop_back() const {
        assert(!empty());
        return erase(size() - 1);
    }

    PersistentDynamicArray pop_front() const {
        assert(!empty());
        return erase(0);
    }

    const T& at(int pos) const {
        assert(0 <= pos && pos < size());
        return (*pool)[find_node(root, pos)].val;
    }

    const T& operator[](int pos) const {
        return at(pos);
    }

    const T& front() const {
        assert(!empty());
        return at(0);
    }

    const T& back() const {
        assert(!empty());
        return at(size() - 1);
    }

    T get(int pos) const {
        return at(pos);
    }

    PersistentDynamicArray set(int pos, T val) const {
        assert(0 <= pos && pos < size());
        return make_version(set_node(root, pos, std::move(val)), rng_state);
    }

    void set_inplace(int pos, T val) {
        assert(0 <= pos && pos < size());
        int next_root = set_node_inplace(root, pos, std::move(val));
        pool->replace(root, next_root);
        pool->discard_unreferenced();
    }

    PersistentDynamicArray reverse(int l, int r) const {
        assert(0 <= l && l <= r && r <= size());
        if (l == r) return *this;
        auto [a, b] = split_node(root, l);
        auto [mid, c] = split_node(b, r - l);
        return make_version(merge(merge(a, reversed_node(mid)), c), rng_state);
    }

    PersistentDynamicArray reverse() const {
        return make_version(reversed_node(root), rng_state);
    }

    PersistentDynamicArray rotate(int l, int m, int r) const {
        assert(0 <= l && l <= m && m <= r && r <= size());
        if (l == m || m == r) return *this;
        auto [a, b] = split_node(root, l);
        auto [c, d] = split_node(b, m - l);
        auto [e, f] = split_node(d, r - m);
        return make_version(merge(merge(a, e), merge(c, f)), rng_state);
    }

    std::pair<PersistentDynamicArray, PersistentDynamicArray> split(int pos) const {
        assert(0 <= pos && pos <= size());
        auto [l, r] = split_node(root, pos);
        PersistentDynamicArray left(l, rng_state, pool);
        PersistentDynamicArray right(r, rng_state, pool);
        pool->discard_unreferenced();
        return {std::move(left), std::move(right)};
    }

    PersistentDynamicArray split_off(int pos) const {
        assert(0 <= pos && pos <= size());
        return make_version(split_node(root, pos).second, rng_state);
    }

    std::vector<T> to_vector() const {
        std::vector<T> res;
        res.reserve(size());
        dump_dfs(root, res);
        return res;
    }

    std::vector<T> to_vector(int l, int r) const {
        assert(0 <= l && l <= r && r <= size());
        std::vector<T> res;
        res.reserve(r - l);
        dump_range_dfs(root, l, r, 0, res);
        return res;
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "ds/dynamic_array/persistent_dynamic_lazy_monoid_array.hpp"



#line 13 "ds/dynamic_array/persistent_dynamic_lazy_monoid_array.hpp"

#line 1 "acted_monoid/concept.hpp"



#line 5 "acted_monoid/concept.hpp"

namespace m1une {
namespace acted_monoid {

// Concept defining the requirements for an Acted Monoid.
template <typename AM>
concept IsActedMonoid = requires(typename AM::value_type a, typename AM::value_type b, typename AM::operator_type f,
                                 typename AM::operator_type g) {
    // 1. Value Monoid
    typename AM::value_type;
    { AM::id() } -> std::same_as<typename AM::value_type>;
    { AM::op(a, b) } -> std::same_as<typename AM::value_type>;

    // 2. Operator Monoid
    typename AM::operator_type;
    { AM::op_id() } -> std::same_as<typename AM::operator_type>;
    { AM::op_comp(f, g) } -> std::same_as<typename AM::operator_type>;  // Composition order: f(g(x))

    // 3. Mapping: Operator x Value -> Value
    { AM::mapping(f, a) } -> std::same_as<typename AM::value_type>;
};

// Concept for acted monoids whose value monoid is a commutative group.
// The value operation must obey commutativity and inverse laws.
template <typename AM>
concept IsCommutativeActedGroup = IsActedMonoid<AM> && requires(typename AM::value_type a) {
    { AM::inv(a) } -> std::same_as<typename AM::value_type>;
};

}  // namespace acted_monoid
}  // namespace m1une


#line 16 "ds/dynamic_array/persistent_dynamic_lazy_monoid_array.hpp"

namespace m1une {
namespace ds {

template <m1une::acted_monoid::IsActedMonoid ActedMonoid>
struct PersistentDynamicLazyMonoidArray {
    using T = typename ActedMonoid::value_type;
    using F = typename ActedMonoid::operator_type;

   private:
    struct Node {
        T val, prod, rprod;
        F lazy;
        int priority;
        int count;
        int l, r;
        bool rev;
        bool has_lazy;

        Node(T value, T product, T reverse_product, F lazy_value, int node_priority, int node_count, int left,
             int right, bool reversed, bool lazy_flag)
            : val(std::move(value)),
              prod(std::move(product)),
              rprod(std::move(reverse_product)),
              lazy(std::move(lazy_value)),
              priority(node_priority),
              count(node_count),
              l(left),
              r(right),
              rev(reversed),
              has_lazy(lazy_flag) {}
    };

    struct BuildNode {
        T val;
        int priority;
        int l, r;

        BuildNode(T value, int node_priority) : val(std::move(value)), priority(node_priority), l(-1), r(-1) {}
    };

    int root;
    std::uint32_t rng_state;
    using Pool = detail::PersistentBinaryNodePool<Node>;

    std::shared_ptr<Pool> pool;

    int subtree_size(int t) const {
        return t == -1 ? 0 : (*pool)[t].count;
    }

    T node_prod(int t) const {
        return t == -1 ? ActedMonoid::id() : (*pool)[t].prod;
    }

    T node_rprod(int t) const {
        return t == -1 ? ActedMonoid::id() : (*pool)[t].rprod;
    }

    static std::uint32_t next_state(std::uint32_t state) {
        state ^= state << 13;
        state ^= state >> 17;
        state ^= state << 5;
        return state == 0 ? 1 : state;
    }

    static int next_priority(std::uint32_t& state) {
        state = next_state(state);
        return int(state);
    }

    template <typename U>
    static T make_value(const U& value) {
        if constexpr (requires(U x) { ActedMonoid::make(x); }) {
            return ActedMonoid::make(value);
        } else {
            return static_cast<T>(value);
        }
    }

    static T mapping_at(const F& f, const T& value, long long ord) {
        if constexpr (requires(F g, T x, long long i) { ActedMonoid::mapping(g, x, i); }) {
            return ActedMonoid::mapping(f, value, ord);
        } else {
            return ActedMonoid::mapping(f, value);
        }
    }

    static F shift_operator(const F& f, long long ord) {
        if constexpr (requires(F g, long long i) { ActedMonoid::op_shift(g, i); }) {
            return ActedMonoid::op_shift(f, ord);
        } else {
            return f;
        }
    }

    static F reverse_operator(const F& f, long long size) {
        if constexpr (requires(F g, long long n) { ActedMonoid::op_reverse(g, n); }) {
            return ActedMonoid::op_reverse(f, size);
        } else {
            return f;
        }
    }

    F compose_for_child(const F& inherited, int t, long long ord) const {
        F shifted = shift_operator(inherited, ord);
        const Node& node = (*pool)[t];
        if (!node.has_lazy) return shifted;
        return ActedMonoid::op_comp(shifted, shift_operator(node.lazy, ord));
    }

    int make_raw_node(T val, T prod, T rprod, F lazy, int priority, int count, bool rev, bool has_lazy, int l,
                      int r) const {
        return pool->emplace(std::move(val), std::move(prod), std::move(rprod), std::move(lazy), priority, count,
                             l, r, rev, has_lazy);
    }

    int make_node(T val, int priority, bool rev, int l, int r) const {
        T prod = ActedMonoid::op(ActedMonoid::op(node_prod(l), val), node_prod(r));
        T rprod = ActedMonoid::op(ActedMonoid::op(node_rprod(r), val), node_rprod(l));
        if (rev) std::swap(prod, rprod);
        int count = 1 + subtree_size(l) + subtree_size(r);
        return make_raw_node(std::move(val), std::move(prod), std::move(rprod), ActedMonoid::op_id(), priority,
                             count, rev, false, l, r);
    }

    int reversed_node(int t) const {
        if (t == -1) return -1;
        Node node = (*pool)[t];
        F lazy = node.has_lazy ? reverse_operator(node.lazy, node.count) : node.lazy;
        return make_raw_node(std::move(node.val), std::move(node.rprod), std::move(node.prod), std::move(lazy),
                             node.priority, node.count, !node.rev, node.has_lazy, node.l, node.r);
    }

    void all_apply_to_node(int t, const F& f) const {
        Node& node = (*pool)[t];
        int left_count = node.rev ? subtree_size(node.r) : subtree_size(node.l);
        node.val = mapping_at(f, node.val, left_count);
        node.prod = mapping_at(f, node.prod, 0);
        node.rprod = mapping_at(reverse_operator(f, node.count), node.rprod, 0);
        node.lazy = ActedMonoid::op_comp(f, node.lazy);
        node.has_lazy = true;
    }

    int all_apply(int t, const F& f) const {
        if (t == -1) return -1;
        int result = pool->clone(t);
        all_apply_to_node(result, f);
        return result;
    }

    int push(int t) const {
        if (t == -1) return -1;
        const Node& stored = (*pool)[t];
        if (!stored.rev && !stored.has_lazy) return t;
        Node node = stored;
        int l = node.l;
        int r = node.r;
        if (node.rev) {
            std::swap(l, r);
            l = reversed_node(l);
            r = reversed_node(r);
        }
        if (node.has_lazy) {
            l = all_apply(l, node.lazy);
            r = all_apply(r, shift_operator(node.lazy, subtree_size(l) + 1));
        }
        return make_node(std::move(node.val), node.priority, false, l, r);
    }

    int merge(int l, int r) const {
        if (l == -1 || r == -1) return l == -1 ? r : l;
        if ((*pool)[l].priority > (*pool)[r].priority) {
            Node node = (*pool)[push(l)];
            int right = merge(node.r, r);
            return make_node(std::move(node.val), node.priority, false, node.l, right);
        }
        Node node = (*pool)[push(r)];
        int left = merge(l, node.l);
        return make_node(std::move(node.val), node.priority, false, left, node.r);
    }

    std::pair<int, int> split_node(int t, int pos) const {
        if (t == -1) return {-1, -1};
        Node node = (*pool)[push(t)];
        int left_count = subtree_size(node.l);
        if (pos <= left_count) {
            auto [a, b] = split_node(node.l, pos);
            return {a, make_node(std::move(node.val), node.priority, false, b, node.r)};
        }
        auto [a, b] = split_node(node.r, pos - left_count - 1);
        return {make_node(std::move(node.val), node.priority, false, node.l, a), b};
    }

    int set_node(int t, int pos, T val) const {
        Node node = (*pool)[push(t)];
        int left_count = subtree_size(node.l);
        if (pos < left_count) {
            int l = set_node(node.l, pos, std::move(val));
            return make_node(std::move(node.val), node.priority, false, l, node.r);
        }
        if (pos == left_count) {
            return make_node(std::move(val), node.priority, false, node.l, node.r);
        }
        int r = set_node(node.r, pos - left_count - 1, std::move(val));
        return make_node(std::move(node.val), node.priority, false, node.l, r);
    }

    void reverse_node_inplace(int t) const {
        Node& node = (*pool)[t];
        std::swap(node.prod, node.rprod);
        if (node.has_lazy) node.lazy = reverse_operator(node.lazy, node.count);
        node.rev = !node.rev;
    }

    int reverse_node_cow(int t) const {
        if (t == -1) return -1;
        t = pool->clone_if_shared(t);
        reverse_node_inplace(t);
        return t;
    }

    int all_apply_cow(int t, const F& f) const {
        if (t == -1) return -1;
        t = pool->clone_if_shared(t);
        all_apply_to_node(t, f);
        return t;
    }

    void pull(int t) const {
        Node& node = (*pool)[t];
        node.prod = ActedMonoid::op(ActedMonoid::op(node_prod(node.l), node.val), node_prod(node.r));
        node.rprod = ActedMonoid::op(ActedMonoid::op(node_rprod(node.r), node.val), node_rprod(node.l));
    }

    void push_inplace(int t) const {
        if (!(*pool)[t].rev && !(*pool)[t].has_lazy) return;
        const bool reversed = (*pool)[t].rev;
        const bool has_lazy = (*pool)[t].has_lazy;
        F lazy = (*pool)[t].lazy;
        int left = (*pool)[t].l;
        int right = (*pool)[t].r;
        if (reversed) {
            int new_left = reverse_node_cow(right);
            int new_right = reverse_node_cow(left);
            // Keep both children alive while the two owning edges are swapped.
            pool->retain(new_left);
            pool->retain(new_right);
            pool->replace((*pool)[t].l, new_left);
            pool->replace((*pool)[t].r, new_right);
            pool->release(new_left);
            pool->release(new_right);
        }
        if (has_lazy) {
            left = all_apply_cow((*pool)[t].l, lazy);
            pool->replace((*pool)[t].l, left);
            right = all_apply_cow((*pool)[t].r, shift_operator(lazy, subtree_size(left) + 1));
            pool->replace((*pool)[t].r, right);
        }
        Node& node = (*pool)[t];
        node.lazy = ActedMonoid::op_id();
        node.rev = false;
        node.has_lazy = false;
        pull(t);
    }

    int set_node_inplace(int t, int pos, T val) const {
        t = pool->clone_if_shared(t);
        push_inplace(t);
        int left_count = subtree_size((*pool)[t].l);
        if (pos < left_count) {
            int child = set_node_inplace((*pool)[t].l, pos, std::move(val));
            pool->replace((*pool)[t].l, child);
        } else if (pos == left_count) {
            (*pool)[t].val = std::move(val);
        } else {
            int child = set_node_inplace((*pool)[t].r, pos - left_count - 1, std::move(val));
            pool->replace((*pool)[t].r, child);
        }
        pull(t);
        return t;
    }

    int apply_node_inplace(int t, int offset, int query_left, int query_right, const F& f) const {
        if (t == -1 || query_right <= offset || offset + subtree_size(t) <= query_left) return t;
        t = pool->clone_if_shared(t);
        if (query_left <= offset && offset + subtree_size(t) <= query_right) {
            all_apply_to_node(t, shift_operator(f, offset - query_left));
            return t;
        }
        push_inplace(t);
        int left_count = subtree_size((*pool)[t].l);
        int child = apply_node_inplace((*pool)[t].l, offset, query_left, query_right, f);
        pool->replace((*pool)[t].l, child);
        int position = offset + left_count;
        if (query_left <= position && position < query_right) {
            (*pool)[t].val = mapping_at(shift_operator(f, position - query_left), (*pool)[t].val, 0);
        }
        child = apply_node_inplace((*pool)[t].r, position + 1, query_left, query_right, f);
        pool->replace((*pool)[t].r, child);
        pull(t);
        return t;
    }

    T get_value(int t, int pos, F inherited, bool reversed = false) const {
        while (t != -1) {
            const Node& node = (*pool)[t];
            bool cur_reversed = reversed ^ node.rev;
            int l = cur_reversed ? node.r : node.l;
            int r = cur_reversed ? node.l : node.r;
            int left_count = subtree_size(l);
            if (pos < left_count) {
                inherited = compose_for_child(inherited, t, 0);
                t = l;
                reversed = cur_reversed;
            } else if (pos == left_count) {
                return mapping_at(inherited, node.val, left_count);
            } else {
                pos -= left_count + 1;
                inherited = compose_for_child(inherited, t, left_count + 1);
                t = r;
                reversed = cur_reversed;
            }
        }
        return ActedMonoid::id();
    }

    T prod_dfs(int t, int ql, int qr, int offset, const F& inherited, bool reversed = false) const {
        if (t == -1 || qr <= offset || offset + (*pool)[t].count <= ql) return ActedMonoid::id();
        const Node& node = (*pool)[t];
        bool cur_reversed = reversed ^ node.rev;
        if (ql <= offset && offset + node.count <= qr) {
            return mapping_at(inherited, reversed ? node.rprod : node.prod, 0);
        }
        int l = cur_reversed ? node.r : node.l;
        int r = cur_reversed ? node.l : node.r;
        int left_count = subtree_size(l);
        int node_pos = offset + left_count;
        T res = prod_dfs(l, ql, qr, offset, compose_for_child(inherited, t, 0), cur_reversed);
        if (ql <= node_pos && node_pos < qr) res = ActedMonoid::op(res, mapping_at(inherited, node.val, left_count));
        return ActedMonoid::op(
            res, prod_dfs(r, ql, qr, node_pos + 1, compose_for_child(inherited, t, left_count + 1),
                          cur_reversed));
    }

    void dump_dfs(int t, std::vector<T>& res, const F& inherited, bool reversed = false) const {
        if (t == -1) return;
        const Node& node = (*pool)[t];
        bool cur_reversed = reversed ^ node.rev;
        int l = cur_reversed ? node.r : node.l;
        int r = cur_reversed ? node.l : node.r;
        int left_count = subtree_size(l);
        dump_dfs(l, res, compose_for_child(inherited, t, 0), cur_reversed);
        res.push_back(mapping_at(inherited, node.val, left_count));
        dump_dfs(r, res, compose_for_child(inherited, t, left_count + 1), cur_reversed);
    }

    void dump_range_dfs(int t, int ql, int qr, int offset, std::vector<T>& res, const F& inherited,
                        bool reversed = false) const {
        if (t == -1 || qr <= offset || offset + (*pool)[t].count <= ql) return;
        const Node& node = (*pool)[t];
        bool cur_reversed = reversed ^ node.rev;
        int l = cur_reversed ? node.r : node.l;
        int r = cur_reversed ? node.l : node.r;
        int left_count = subtree_size(l);
        int node_pos = offset + left_count;
        dump_range_dfs(l, ql, qr, offset, res, compose_for_child(inherited, t, 0), cur_reversed);
        if (ql <= node_pos && node_pos < qr) res.push_back(mapping_at(inherited, node.val, left_count));
        dump_range_dfs(r, ql, qr, node_pos + 1, res, compose_for_child(inherited, t, left_count + 1),
                       cur_reversed);
    }

    int build_from_nodes(std::vector<BuildNode>& nodes, int t) const {
        if (t == -1) return -1;
        int l = build_from_nodes(nodes, nodes[t].l);
        int r = build_from_nodes(nodes, nodes[t].r);
        return make_node(std::move(nodes[t].val), nodes[t].priority, false, l, r);
    }

    int build_cartesian(std::vector<BuildNode>& nodes) const {
        if (nodes.empty()) return -1;
        std::vector<int> stack;
        stack.reserve(nodes.size());
        for (int i = 0; i < int(nodes.size()); i++) {
            int left_child = -1;
            while (!stack.empty() && nodes[stack.back()].priority < nodes[i].priority) {
                left_child = stack.back();
                stack.pop_back();
            }
            nodes[i].l = left_child;
            if (!stack.empty()) nodes[stack.back()].r = i;
            stack.push_back(i);
        }
        return build_from_nodes(nodes, stack.front());
    }

    int build_from_vector(const std::vector<T>& v, std::uint32_t& state) const {
        std::vector<BuildNode> nodes;
        nodes.reserve(v.size());
        for (const T& x : v) nodes.emplace_back(x, next_priority(state));
        return build_cartesian(nodes);
    }

    int build_from_vector(std::vector<T>&& v, std::uint32_t& state) const {
        std::vector<BuildNode> nodes;
        nodes.reserve(v.size());
        for (T& x : v) nodes.emplace_back(std::move(x), next_priority(state));
        return build_cartesian(nodes);
    }

    template <typename U>
    int build_from_values(const std::vector<U>& v, std::uint32_t& state) const {
        std::vector<BuildNode> nodes;
        nodes.reserve(v.size());
        for (const U& x : v) nodes.emplace_back(make_value(x), next_priority(state));
        return build_cartesian(nodes);
    }

    int import_node(const PersistentDynamicLazyMonoidArray& other, int t) const {
        if (t == -1) return -1;
        if (pool == other.pool) return t;
        const Node& node = (*other.pool)[t];
        int l = import_node(other, node.l);
        int r = import_node(other, node.r);
        return make_raw_node(node.val, node.prod, node.rprod, node.lazy, node.priority, node.count, node.rev,
                             node.has_lazy, l, r);
    }

    explicit PersistentDynamicLazyMonoidArray(int node, std::uint32_t state,
                                              std::shared_ptr<Pool> node_pool)
        : root(node), rng_state(state), pool(std::move(node_pool)) {
        pool->retain(root);
    }

    PersistentDynamicLazyMonoidArray make_version(int node, std::uint32_t state) const {
        PersistentDynamicLazyMonoidArray result(node, state, pool);
        pool->discard_unreferenced();
        return result;
    }

   public:
    PersistentDynamicLazyMonoidArray()
        : root(-1),
          rng_state(std::uint32_t(std::chrono::steady_clock::now().time_since_epoch().count())),
          pool(std::make_shared<Pool>()) {
        if (rng_state == 0) rng_state = 1;
    }

    explicit PersistentDynamicLazyMonoidArray(int n)
        : PersistentDynamicLazyMonoidArray(n, ActedMonoid::id()) {}

    PersistentDynamicLazyMonoidArray(int n, const T& value) : PersistentDynamicLazyMonoidArray() {
        assert(0 <= n);
        pool->reserve(n);
        std::vector<T> v(n, value);
        root = build_from_vector(std::move(v), rng_state);
        pool->retain(root);
        pool->discard_unreferenced();
    }

    explicit PersistentDynamicLazyMonoidArray(const std::vector<T>& v)
        : PersistentDynamicLazyMonoidArray() {
        pool->reserve(v.size());
        root = build_from_vector(v, rng_state);
        pool->retain(root);
        pool->discard_unreferenced();
    }

    explicit PersistentDynamicLazyMonoidArray(std::vector<T>&& v) : PersistentDynamicLazyMonoidArray() {
        pool->reserve(v.size());
        root = build_from_vector(std::move(v), rng_state);
        pool->retain(root);
        pool->discard_unreferenced();
    }

    template <typename U>
        requires(!std::same_as<U, T>) &&
                (requires(U x) { ActedMonoid::make(x); } || std::convertible_to<U, T>)
    explicit PersistentDynamicLazyMonoidArray(const std::vector<U>& v)
        : PersistentDynamicLazyMonoidArray() {
        pool->reserve(v.size());
        root = build_from_values(v, rng_state);
        pool->retain(root);
        pool->discard_unreferenced();
    }

    PersistentDynamicLazyMonoidArray(std::initializer_list<T> init)
        : PersistentDynamicLazyMonoidArray(std::vector<T>(init)) {}

    PersistentDynamicLazyMonoidArray(const PersistentDynamicLazyMonoidArray& other)
        : root(other.root), rng_state(other.rng_state), pool(other.pool) {
        if (pool) pool->retain(root);
    }

    PersistentDynamicLazyMonoidArray(PersistentDynamicLazyMonoidArray&& other) noexcept
        : root(other.root), rng_state(other.rng_state), pool(std::move(other.pool)) {
        other.root = -1;
    }

    PersistentDynamicLazyMonoidArray& operator=(const PersistentDynamicLazyMonoidArray& other) {
        if (this == &other) return *this;
        if (other.pool) other.pool->retain(other.root);
        if (pool) pool->release(root);
        root = other.root;
        rng_state = other.rng_state;
        pool = other.pool;
        return *this;
    }

    PersistentDynamicLazyMonoidArray& operator=(PersistentDynamicLazyMonoidArray&& other) noexcept {
        if (this == &other) return *this;
        if (pool) pool->release(root);
        root = other.root;
        rng_state = other.rng_state;
        pool = std::move(other.pool);
        other.root = -1;
        return *this;
    }

    ~PersistentDynamicLazyMonoidArray() {
        if (pool) pool->release(root);
    }

    int size() const {
        return subtree_size(root);
    }

    bool empty() const {
        return size() == 0;
    }

    void release() {
        if (pool) pool->release(root);
        root = -1;
        pool = std::make_shared<Pool>();
    }

    std::size_t node_count() const { return pool ? pool->size() : 0; }

    PersistentDynamicLazyMonoidArray clear() const {
        return make_version(-1, rng_state);
    }

    PersistentDynamicLazyMonoidArray insert(int pos, T value) const {
        assert(0 <= pos && pos <= size());
        std::uint32_t next = next_state(rng_state);
        int node = make_node(std::move(value), int(next), false, -1, -1);
        auto [l, r] = split_node(root, pos);
        return make_version(merge(merge(l, node), r), next);
    }

    PersistentDynamicLazyMonoidArray insert(int pos, const std::vector<T>& v) const {
        assert(0 <= pos && pos <= size());
        if (v.empty()) return *this;
        std::uint32_t next = rng_state;
        int mid = build_from_vector(v, next);
        auto [l, r] = split_node(root, pos);
        return make_version(merge(merge(l, mid), r), next);
    }

    PersistentDynamicLazyMonoidArray insert(int pos, std::vector<T>&& v) const {
        assert(0 <= pos && pos <= size());
        if (v.empty()) return *this;
        std::uint32_t next = rng_state;
        int mid = build_from_vector(std::move(v), next);
        auto [l, r] = split_node(root, pos);
        return make_version(merge(merge(l, mid), r), next);
    }

    PersistentDynamicLazyMonoidArray insert(int pos, std::initializer_list<T> init) const {
        return insert(pos, std::vector<T>(init));
    }

    PersistentDynamicLazyMonoidArray insert(int pos, const PersistentDynamicLazyMonoidArray& other) const {
        assert(0 <= pos && pos <= size());
        if (other.empty()) return *this;
        int mid = import_node(other, other.root);
        auto [l, r] = split_node(root, pos);
        return make_version(merge(merge(l, mid), r), rng_state);
    }

    PersistentDynamicLazyMonoidArray push_back(T value) const {
        return insert(size(), std::move(value));
    }

    PersistentDynamicLazyMonoidArray push_front(T value) const {
        return insert(0, std::move(value));
    }

    PersistentDynamicLazyMonoidArray append(const std::vector<T>& v) const {
        return insert(size(), v);
    }

    PersistentDynamicLazyMonoidArray append(std::vector<T>&& v) const {
        return insert(size(), std::move(v));
    }

    PersistentDynamicLazyMonoidArray append(const PersistentDynamicLazyMonoidArray& other) const {
        return insert(size(), other);
    }

    PersistentDynamicLazyMonoidArray erase(int pos) const {
        assert(0 <= pos && pos < size());
        auto [a, b] = split_node(root, pos);
        auto [mid, c] = split_node(b, 1);
        (void)mid;
        return make_version(merge(a, c), rng_state);
    }

    PersistentDynamicLazyMonoidArray erase(int l, int r) const {
        assert(0 <= l && l <= r && r <= size());
        if (l == r) return *this;
        auto [a, b] = split_node(root, l);
        auto [mid, c] = split_node(b, r - l);
        (void)mid;
        return make_version(merge(a, c), rng_state);
    }

    PersistentDynamicLazyMonoidArray pop_back() const {
        assert(!empty());
        return erase(size() - 1);
    }

    PersistentDynamicLazyMonoidArray pop_front() const {
        assert(!empty());
        return erase(0);
    }

    T get(int pos) const {
        assert(0 <= pos && pos < size());
        return get_value(root, pos, ActedMonoid::op_id());
    }

    T operator[](int pos) const {
        return get(pos);
    }

    T front() const {
        assert(!empty());
        return get(0);
    }

    T back() const {
        assert(!empty());
        return get(size() - 1);
    }

    PersistentDynamicLazyMonoidArray set(int pos, T value) const {
        assert(0 <= pos && pos < size());
        return make_version(set_node(root, pos, std::move(value)), rng_state);
    }

    void set_inplace(int pos, T value) {
        assert(0 <= pos && pos < size());
        int next_root = set_node_inplace(root, pos, std::move(value));
        pool->replace(root, next_root);
        pool->discard_unreferenced();
    }

    PersistentDynamicLazyMonoidArray reverse(int l, int r) const {
        assert(0 <= l && l <= r && r <= size());
        if (l == r) return *this;
        auto [a, b] = split_node(root, l);
        auto [mid, c] = split_node(b, r - l);
        return make_version(merge(merge(a, reversed_node(mid)), c), rng_state);
    }

    PersistentDynamicLazyMonoidArray reverse() const {
        return make_version(reversed_node(root), rng_state);
    }

    PersistentDynamicLazyMonoidArray rotate(int l, int m, int r) const {
        assert(0 <= l && l <= m && m <= r && r <= size());
        if (l == m || m == r) return *this;
        auto [a, b] = split_node(root, l);
        auto [c, d] = split_node(b, m - l);
        auto [e, f] = split_node(d, r - m);
        return make_version(merge(merge(a, e), merge(c, f)), rng_state);
    }

    PersistentDynamicLazyMonoidArray apply(int pos, const F& f) const {
        assert(0 <= pos && pos < size());
        return apply(pos, pos + 1, f);
    }

    PersistentDynamicLazyMonoidArray apply(int l, int r, const F& f) const {
        assert(0 <= l && l <= r && r <= size());
        if (l == r) return *this;
        auto [a, b] = split_node(root, l);
        auto [mid, c] = split_node(b, r - l);
        return make_version(merge(merge(a, all_apply(mid, f)), c), rng_state);
    }

    void apply_inplace(int pos, const F& f) {
        assert(0 <= pos && pos < size());
        apply_inplace(pos, pos + 1, f);
    }

    void apply_inplace(int l, int r, const F& f) {
        assert(0 <= l && l <= r && r <= size());
        if (l == r) return;
        int next_root = apply_node_inplace(root, 0, l, r, f);
        pool->replace(root, next_root);
        pool->discard_unreferenced();
    }

    T prod(int l, int r) const {
        assert(0 <= l && l <= r && r <= size());
        if (l == r) return ActedMonoid::id();
        return prod_dfs(root, l, r, 0, ActedMonoid::op_id());
    }

    T all_prod() const {
        return root == -1 ? ActedMonoid::id() : (*pool)[root].prod;
    }

    std::pair<PersistentDynamicLazyMonoidArray, PersistentDynamicLazyMonoidArray> split(int pos) const {
        assert(0 <= pos && pos <= size());
        auto [l, r] = split_node(root, pos);
        PersistentDynamicLazyMonoidArray left(l, rng_state, pool);
        PersistentDynamicLazyMonoidArray right(r, rng_state, pool);
        pool->discard_unreferenced();
        return {std::move(left), std::move(right)};
    }

    PersistentDynamicLazyMonoidArray split_off(int pos) const {
        assert(0 <= pos && pos <= size());
        return make_version(split_node(root, pos).second, rng_state);
    }

    std::vector<T> to_vector() const {
        std::vector<T> res;
        res.reserve(size());
        dump_dfs(root, res, ActedMonoid::op_id());
        return res;
    }

    std::vector<T> to_vector(int l, int r) const {
        assert(0 <= l && l <= r && r <= size());
        std::vector<T> res;
        res.reserve(r - l);
        dump_range_dfs(root, l, r, 0, res, ActedMonoid::op_id());
        return res;
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "ds/dynamic_array/persistent_dynamic_monoid_array.hpp"



#line 13 "ds/dynamic_array/persistent_dynamic_monoid_array.hpp"

#line 16 "ds/dynamic_array/persistent_dynamic_monoid_array.hpp"

namespace m1une {
namespace ds {

template <m1une::monoid::IsMonoid Monoid>
struct PersistentDynamicMonoidArray {
    using T = typename Monoid::value_type;

   private:
    struct Node {
        T val, prod, rprod;
        int priority;
        int count;
        int l, r;
        bool rev;

        Node(T value, T product, T reverse_product, int node_priority, int node_count, int left, int right,
             bool reversed)
            : val(std::move(value)),
              prod(std::move(product)),
              rprod(std::move(reverse_product)),
              priority(node_priority),
              count(node_count),
              l(left),
              r(right),
              rev(reversed) {}
    };

    struct BuildNode {
        T val;
        int priority;
        int l, r;

        BuildNode(T value, int node_priority) : val(std::move(value)), priority(node_priority), l(-1), r(-1) {}
    };

    int root;
    std::uint32_t rng_state;
    using Pool = detail::PersistentBinaryNodePool<Node>;

    std::shared_ptr<Pool> pool;

    int subtree_size(int t) const {
        return t == -1 ? 0 : (*pool)[t].count;
    }

    T node_prod(int t) const {
        return t == -1 ? Monoid::id() : (*pool)[t].prod;
    }

    T node_rprod(int t) const {
        return t == -1 ? Monoid::id() : (*pool)[t].rprod;
    }

    static std::uint32_t next_state(std::uint32_t state) {
        state ^= state << 13;
        state ^= state >> 17;
        state ^= state << 5;
        return state == 0 ? 1 : state;
    }

    static int next_priority(std::uint32_t& state) {
        state = next_state(state);
        return int(state);
    }

    template <typename U>
    static T make_value(const U& value) {
        if constexpr (requires(U x) { Monoid::make(x); }) {
            return Monoid::make(value);
        } else {
            return static_cast<T>(value);
        }
    }

    int make_node(T val, int priority, bool rev, int l, int r) const {
        T prod = Monoid::op(Monoid::op(node_prod(l), val), node_prod(r));
        T rprod = Monoid::op(Monoid::op(node_rprod(r), val), node_rprod(l));
        if (rev) std::swap(prod, rprod);
        int count = 1 + subtree_size(l) + subtree_size(r);
        return pool->emplace(std::move(val), std::move(prod), std::move(rprod), priority, count, l, r, rev);
    }

    int reversed_node(int t) const {
        if (t == -1) return -1;
        Node node = (*pool)[t];
        return make_node(std::move(node.val), node.priority, !node.rev, node.l, node.r);
    }

    int push(int t) const {
        if (t == -1 || !(*pool)[t].rev) return t;
        Node node = (*pool)[t];
        int l = reversed_node(node.r);
        int r = reversed_node(node.l);
        return make_node(std::move(node.val), node.priority, false, l, r);
    }

    int merge(int l, int r) const {
        if (l == -1 || r == -1) return l == -1 ? r : l;
        if ((*pool)[l].priority > (*pool)[r].priority) {
            Node node = (*pool)[push(l)];
            int right = merge(node.r, r);
            return make_node(std::move(node.val), node.priority, false, node.l, right);
        }
        Node node = (*pool)[push(r)];
        int left = merge(l, node.l);
        return make_node(std::move(node.val), node.priority, false, left, node.r);
    }

    std::pair<int, int> split_node(int t, int pos) const {
        if (t == -1) return {-1, -1};
        Node node = (*pool)[push(t)];
        int left_count = subtree_size(node.l);
        if (pos <= left_count) {
            auto [a, b] = split_node(node.l, pos);
            return {a, make_node(std::move(node.val), node.priority, false, b, node.r)};
        }
        auto [a, b] = split_node(node.r, pos - left_count - 1);
        return {make_node(std::move(node.val), node.priority, false, node.l, a), b};
    }

    int set_node(int t, int pos, T val) const {
        Node node = (*pool)[push(t)];
        int left_count = subtree_size(node.l);
        if (pos < left_count) {
            int l = set_node(node.l, pos, std::move(val));
            return make_node(std::move(node.val), node.priority, false, l, node.r);
        }
        if (pos == left_count) {
            return make_node(std::move(val), node.priority, false, node.l, node.r);
        }
        int r = set_node(node.r, pos - left_count - 1, std::move(val));
        return make_node(std::move(node.val), node.priority, false, node.l, r);
    }

    void pull(int t) const {
        Node& node = (*pool)[t];
        node.prod = Monoid::op(Monoid::op(node_prod(node.l), node.val), node_prod(node.r));
        node.rprod = Monoid::op(Monoid::op(node_rprod(node.r), node.val), node_rprod(node.l));
        if (node.rev) std::swap(node.prod, node.rprod);
    }

    int set_node_inplace(int t, int pos, T val, bool inherited_reversed = false) const {
        t = pool->clone_if_shared(t);
        const bool reversed = inherited_reversed ^ (*pool)[t].rev;
        const int logical_left = reversed ? (*pool)[t].r : (*pool)[t].l;
        const int left_count = subtree_size(logical_left);
        if (pos < left_count) {
            int child = set_node_inplace(logical_left, pos, std::move(val), reversed);
            if (reversed) {
                pool->replace((*pool)[t].r, child);
            } else {
                pool->replace((*pool)[t].l, child);
            }
        } else if (pos == left_count) {
            (*pool)[t].val = std::move(val);
        } else {
            const int logical_right = reversed ? (*pool)[t].l : (*pool)[t].r;
            int child = set_node_inplace(logical_right, pos - left_count - 1, std::move(val), reversed);
            if (reversed) {
                pool->replace((*pool)[t].l, child);
            } else {
                pool->replace((*pool)[t].r, child);
            }
        }
        pull(t);
        return t;
    }

    int find_node(int t, int pos) const {
        bool reversed = false;
        while (t != -1) {
            const Node& node = (*pool)[t];
            bool cur_reversed = reversed ^ node.rev;
            int l = cur_reversed ? node.r : node.l;
            int r = cur_reversed ? node.l : node.r;
            int left_count = subtree_size(l);
            if (pos < left_count) {
                t = l;
                reversed = cur_reversed;
            } else if (pos == left_count) {
                return t;
            } else {
                pos -= left_count + 1;
                t = r;
                reversed = cur_reversed;
            }
        }
        return -1;
    }

    T prod_dfs(int t, int ql, int qr, int offset, bool reversed = false) const {
        if (t == -1 || qr <= offset || offset + (*pool)[t].count <= ql) return Monoid::id();
        const Node& node = (*pool)[t];
        if (ql <= offset && offset + node.count <= qr) return reversed ? node.rprod : node.prod;
        bool cur_reversed = reversed ^ node.rev;
        int l = cur_reversed ? node.r : node.l;
        int r = cur_reversed ? node.l : node.r;
        int left_count = subtree_size(l);
        int node_pos = offset + left_count;
        T res = prod_dfs(l, ql, qr, offset, cur_reversed);
        if (ql <= node_pos && node_pos < qr) res = Monoid::op(res, node.val);
        return Monoid::op(res, prod_dfs(r, ql, qr, node_pos + 1, cur_reversed));
    }

    void dump_dfs(int t, std::vector<T>& res, bool reversed = false) const {
        if (t == -1) return;
        const Node& node = (*pool)[t];
        bool cur_reversed = reversed ^ node.rev;
        int l = cur_reversed ? node.r : node.l;
        int r = cur_reversed ? node.l : node.r;
        dump_dfs(l, res, cur_reversed);
        res.push_back(node.val);
        dump_dfs(r, res, cur_reversed);
    }

    void dump_range_dfs(int t, int ql, int qr, int offset, std::vector<T>& res, bool reversed = false) const {
        if (t == -1 || qr <= offset || offset + (*pool)[t].count <= ql) return;
        const Node& node = (*pool)[t];
        bool cur_reversed = reversed ^ node.rev;
        int l = cur_reversed ? node.r : node.l;
        int r = cur_reversed ? node.l : node.r;
        int left_count = subtree_size(l);
        int node_pos = offset + left_count;
        dump_range_dfs(l, ql, qr, offset, res, cur_reversed);
        if (ql <= node_pos && node_pos < qr) res.push_back(node.val);
        dump_range_dfs(r, ql, qr, node_pos + 1, res, cur_reversed);
    }

    int build_from_nodes(std::vector<BuildNode>& nodes, int t) const {
        if (t == -1) return -1;
        int l = build_from_nodes(nodes, nodes[t].l);
        int r = build_from_nodes(nodes, nodes[t].r);
        return make_node(std::move(nodes[t].val), nodes[t].priority, false, l, r);
    }

    int build_cartesian(std::vector<BuildNode>& nodes) const {
        if (nodes.empty()) return -1;
        std::vector<int> stack;
        stack.reserve(nodes.size());
        for (int i = 0; i < int(nodes.size()); i++) {
            int left_child = -1;
            while (!stack.empty() && nodes[stack.back()].priority < nodes[i].priority) {
                left_child = stack.back();
                stack.pop_back();
            }
            nodes[i].l = left_child;
            if (!stack.empty()) nodes[stack.back()].r = i;
            stack.push_back(i);
        }
        return build_from_nodes(nodes, stack.front());
    }

    int build_from_vector(const std::vector<T>& v, std::uint32_t& state) const {
        std::vector<BuildNode> nodes;
        nodes.reserve(v.size());
        for (const T& x : v) nodes.emplace_back(x, next_priority(state));
        return build_cartesian(nodes);
    }

    int build_from_vector(std::vector<T>&& v, std::uint32_t& state) const {
        std::vector<BuildNode> nodes;
        nodes.reserve(v.size());
        for (T& x : v) nodes.emplace_back(std::move(x), next_priority(state));
        return build_cartesian(nodes);
    }

    template <typename U>
    int build_from_values(const std::vector<U>& v, std::uint32_t& state) const {
        std::vector<BuildNode> nodes;
        nodes.reserve(v.size());
        for (const U& x : v) nodes.emplace_back(make_value(x), next_priority(state));
        return build_cartesian(nodes);
    }

    int import_node(const PersistentDynamicMonoidArray& other, int t) const {
        if (t == -1) return -1;
        if (pool == other.pool) return t;
        const Node& node = (*other.pool)[t];
        int l = import_node(other, node.l);
        int r = import_node(other, node.r);
        return make_node(node.val, node.priority, node.rev, l, r);
    }

    explicit PersistentDynamicMonoidArray(int node, std::uint32_t state,
                                          std::shared_ptr<Pool> node_pool)
        : root(node), rng_state(state), pool(std::move(node_pool)) {
        pool->retain(root);
    }

    PersistentDynamicMonoidArray make_version(int node, std::uint32_t state) const {
        PersistentDynamicMonoidArray result(node, state, pool);
        pool->discard_unreferenced();
        return result;
    }

   public:
    PersistentDynamicMonoidArray()
        : root(-1),
          rng_state(std::uint32_t(std::chrono::steady_clock::now().time_since_epoch().count())),
          pool(std::make_shared<Pool>()) {
        if (rng_state == 0) rng_state = 1;
    }

    explicit PersistentDynamicMonoidArray(int n) : PersistentDynamicMonoidArray(n, Monoid::id()) {}

    PersistentDynamicMonoidArray(int n, const T& value) : PersistentDynamicMonoidArray() {
        assert(0 <= n);
        pool->reserve(n);
        std::vector<T> v(n, value);
        root = build_from_vector(std::move(v), rng_state);
        pool->retain(root);
        pool->discard_unreferenced();
    }

    explicit PersistentDynamicMonoidArray(const std::vector<T>& v) : PersistentDynamicMonoidArray() {
        pool->reserve(v.size());
        root = build_from_vector(v, rng_state);
        pool->retain(root);
        pool->discard_unreferenced();
    }

    explicit PersistentDynamicMonoidArray(std::vector<T>&& v) : PersistentDynamicMonoidArray() {
        pool->reserve(v.size());
        root = build_from_vector(std::move(v), rng_state);
        pool->retain(root);
        pool->discard_unreferenced();
    }

    template <typename U>
        requires(!std::same_as<U, T>) && (requires(U x) { Monoid::make(x); } || std::convertible_to<U, T>)
    explicit PersistentDynamicMonoidArray(const std::vector<U>& v) : PersistentDynamicMonoidArray() {
        pool->reserve(v.size());
        root = build_from_values(v, rng_state);
        pool->retain(root);
        pool->discard_unreferenced();
    }

    PersistentDynamicMonoidArray(std::initializer_list<T> init)
        : PersistentDynamicMonoidArray(std::vector<T>(init)) {}

    PersistentDynamicMonoidArray(const PersistentDynamicMonoidArray& other)
        : root(other.root), rng_state(other.rng_state), pool(other.pool) {
        if (pool) pool->retain(root);
    }

    PersistentDynamicMonoidArray(PersistentDynamicMonoidArray&& other) noexcept
        : root(other.root), rng_state(other.rng_state), pool(std::move(other.pool)) {
        other.root = -1;
    }

    PersistentDynamicMonoidArray& operator=(const PersistentDynamicMonoidArray& other) {
        if (this == &other) return *this;
        if (other.pool) other.pool->retain(other.root);
        if (pool) pool->release(root);
        root = other.root;
        rng_state = other.rng_state;
        pool = other.pool;
        return *this;
    }

    PersistentDynamicMonoidArray& operator=(PersistentDynamicMonoidArray&& other) noexcept {
        if (this == &other) return *this;
        if (pool) pool->release(root);
        root = other.root;
        rng_state = other.rng_state;
        pool = std::move(other.pool);
        other.root = -1;
        return *this;
    }

    ~PersistentDynamicMonoidArray() {
        if (pool) pool->release(root);
    }

    int size() const {
        return subtree_size(root);
    }

    bool empty() const {
        return size() == 0;
    }

    void release() {
        if (pool) pool->release(root);
        root = -1;
        pool = std::make_shared<Pool>();
    }

    std::size_t node_count() const { return pool ? pool->size() : 0; }

    PersistentDynamicMonoidArray clear() const {
        return make_version(-1, rng_state);
    }

    PersistentDynamicMonoidArray insert(int pos, T value) const {
        assert(0 <= pos && pos <= size());
        std::uint32_t next = next_state(rng_state);
        int node = make_node(std::move(value), int(next), false, -1, -1);
        auto [l, r] = split_node(root, pos);
        return make_version(merge(merge(l, node), r), next);
    }

    PersistentDynamicMonoidArray insert(int pos, const std::vector<T>& v) const {
        assert(0 <= pos && pos <= size());
        if (v.empty()) return *this;
        std::uint32_t next = rng_state;
        int mid = build_from_vector(v, next);
        auto [l, r] = split_node(root, pos);
        return make_version(merge(merge(l, mid), r), next);
    }

    PersistentDynamicMonoidArray insert(int pos, std::vector<T>&& v) const {
        assert(0 <= pos && pos <= size());
        if (v.empty()) return *this;
        std::uint32_t next = rng_state;
        int mid = build_from_vector(std::move(v), next);
        auto [l, r] = split_node(root, pos);
        return make_version(merge(merge(l, mid), r), next);
    }

    PersistentDynamicMonoidArray insert(int pos, std::initializer_list<T> init) const {
        return insert(pos, std::vector<T>(init));
    }

    PersistentDynamicMonoidArray insert(int pos, const PersistentDynamicMonoidArray& other) const {
        assert(0 <= pos && pos <= size());
        if (other.empty()) return *this;
        int mid = import_node(other, other.root);
        auto [l, r] = split_node(root, pos);
        return make_version(merge(merge(l, mid), r), rng_state);
    }

    PersistentDynamicMonoidArray push_back(T value) const {
        return insert(size(), std::move(value));
    }

    PersistentDynamicMonoidArray push_front(T value) const {
        return insert(0, std::move(value));
    }

    PersistentDynamicMonoidArray append(const std::vector<T>& v) const {
        return insert(size(), v);
    }

    PersistentDynamicMonoidArray append(std::vector<T>&& v) const {
        return insert(size(), std::move(v));
    }

    PersistentDynamicMonoidArray append(const PersistentDynamicMonoidArray& other) const {
        return insert(size(), other);
    }

    PersistentDynamicMonoidArray erase(int pos) const {
        assert(0 <= pos && pos < size());
        auto [a, b] = split_node(root, pos);
        auto [mid, c] = split_node(b, 1);
        (void)mid;
        return make_version(merge(a, c), rng_state);
    }

    PersistentDynamicMonoidArray erase(int l, int r) const {
        assert(0 <= l && l <= r && r <= size());
        if (l == r) return *this;
        auto [a, b] = split_node(root, l);
        auto [mid, c] = split_node(b, r - l);
        (void)mid;
        return make_version(merge(a, c), rng_state);
    }

    PersistentDynamicMonoidArray pop_back() const {
        assert(!empty());
        return erase(size() - 1);
    }

    PersistentDynamicMonoidArray pop_front() const {
        assert(!empty());
        return erase(0);
    }

    T get(int pos) const {
        assert(0 <= pos && pos < size());
        return (*pool)[find_node(root, pos)].val;
    }

    T operator[](int pos) const {
        return get(pos);
    }

    T front() const {
        assert(!empty());
        return get(0);
    }

    T back() const {
        assert(!empty());
        return get(size() - 1);
    }

    PersistentDynamicMonoidArray set(int pos, T value) const {
        assert(0 <= pos && pos < size());
        return make_version(set_node(root, pos, std::move(value)), rng_state);
    }

    void set_inplace(int pos, T value) {
        assert(0 <= pos && pos < size());
        int next_root = set_node_inplace(root, pos, std::move(value));
        pool->replace(root, next_root);
        pool->discard_unreferenced();
    }

    PersistentDynamicMonoidArray reverse(int l, int r) const {
        assert(0 <= l && l <= r && r <= size());
        if (l == r) return *this;
        auto [a, b] = split_node(root, l);
        auto [mid, c] = split_node(b, r - l);
        return make_version(merge(merge(a, reversed_node(mid)), c), rng_state);
    }

    PersistentDynamicMonoidArray reverse() const {
        return make_version(reversed_node(root), rng_state);
    }

    PersistentDynamicMonoidArray rotate(int l, int m, int r) const {
        assert(0 <= l && l <= m && m <= r && r <= size());
        if (l == m || m == r) return *this;
        auto [a, b] = split_node(root, l);
        auto [c, d] = split_node(b, m - l);
        auto [e, f] = split_node(d, r - m);
        return make_version(merge(merge(a, e), merge(c, f)), rng_state);
    }

    T prod(int l, int r) const {
        assert(0 <= l && l <= r && r <= size());
        if (l == r) return Monoid::id();
        return prod_dfs(root, l, r, 0);
    }

    T all_prod() const {
        return root == -1 ? Monoid::id() : (*pool)[root].prod;
    }

    std::pair<PersistentDynamicMonoidArray, PersistentDynamicMonoidArray> split(int pos) const {
        assert(0 <= pos && pos <= size());
        auto [l, r] = split_node(root, pos);
        PersistentDynamicMonoidArray left(l, rng_state, pool);
        PersistentDynamicMonoidArray right(r, rng_state, pool);
        pool->discard_unreferenced();
        return {std::move(left), std::move(right)};
    }

    PersistentDynamicMonoidArray split_off(int pos) const {
        assert(0 <= pos && pos <= size());
        return make_version(split_node(root, pos).second, rng_state);
    }

    std::vector<T> to_vector() const {
        std::vector<T> res;
        res.reserve(size());
        dump_dfs(root, res);
        return res;
    }

    std::vector<T> to_vector(int l, int r) const {
        assert(0 <= l && l <= r && r <= size());
        std::vector<T> res;
        res.reserve(r - l);
        dump_range_dfs(root, l, r, 0, res);
        return res;
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "ds/segtree/persistent_dual_segtree.hpp"



#line 9 "ds/segtree/persistent_dual_segtree.hpp"

#line 1 "ds/segtree/persistent_node_pool.hpp"



#line 9 "ds/segtree/persistent_node_pool.hpp"

namespace m1une {
namespace ds {
namespace detail {

// Node must have integer `left`, `right`, and `references` members.
template <class Node>
struct PersistentNodePool {
    std::vector<Node> nodes;
    int first_free = 0;
    std::size_t live_nodes = 0;

   private:
    void release_zero(int node) {
        int left = nodes[node].left;
        int right = nodes[node].right;
        nodes[node] = Node();
        nodes[node].left = first_free;
        first_free = node;
        --live_nodes;
        if (left && --nodes[left].references == 0) release_zero(left);
        if (right && --nodes[right].references == 0) release_zero(right);
    }

   public:
    PersistentNodePool() { nodes.emplace_back(); }

    void reserve(std::size_t capacity) { nodes.reserve(capacity + 1); }

    Node& operator[](int node) { return nodes[node]; }

    const Node& operator[](int node) const { return nodes[node]; }

    void retain(int node) {
        if (node) ++nodes[node].references;
    }

    void release(int node) {
        if (!node) return;
        assert(nodes[node].references > 0);
        if (--nodes[node].references == 0) release_zero(node);
    }

    template <class... Args>
    int emplace(Args&&... args) {
        int result;
        if (!first_free) {
            assert(nodes.size() < std::size_t(std::numeric_limits<int>::max()));
            nodes.emplace_back(std::forward<Args>(args)...);
            result = int(nodes.size()) - 1;
        } else {
            result = first_free;
            first_free = nodes[result].left;
            nodes[result] = Node(std::forward<Args>(args)...);
        }
        Node& node = nodes[result];
        node.references = 0;
        retain(node.left);
        retain(node.right);
        ++live_nodes;
        return result;
    }

    int clone(int node) {
        assert(node);
        Node copy = nodes[node];
        return emplace(std::move(copy));
    }

    bool unique(int node) const {
        return !node || nodes[node].references == 1;
    }

    // Returns node itself when it has one owner, otherwise an unowned clone.
    // The caller must attach a returned clone with replace() before it can be
    // released or exposed as a root.
    int clone_if_shared(int node) {
        if (unique(node)) return node;
        return clone(node);
    }

    void replace(int& edge, int node) {
        if (edge == node) return;
        retain(node);
        int old = edge;
        edge = node;
        release(old);
    }

    std::size_t size() const { return live_nodes; }
};

}  // namespace detail
}  // namespace ds
}  // namespace m1une


#line 12 "ds/segtree/persistent_dual_segtree.hpp"

namespace m1une {
namespace ds {

template <m1une::monoid::IsMonoid Monoid>
struct PersistentDualSegtree {
    using T = typename Monoid::value_type;

   private:
    struct Node {
        T val;
        int left, right;
        int references;
        bool has_lazy;

        Node() : val(Monoid::id()), left(0), right(0), references(0), has_lazy(false) {}
        explicit Node(T value) : val(std::move(value)), left(0), right(0), references(0), has_lazy(false) {}
        Node(int left_child, int right_child)
            : val(Monoid::id()), left(left_child), right(right_child), references(0), has_lazy(false) {}
    };

    using Pool = detail::PersistentNodePool<Node>;

    int _n;
    int _root;
    std::shared_ptr<Pool> _pool;

    explicit PersistentDualSegtree(int n, int root, std::shared_ptr<Pool> pool)
        : _n(n), _root(root), _pool(std::move(pool)) {
        _pool->retain(_root);
    }

    int new_node(const Node& node) const { return _pool->emplace(node); }

    int new_node(Node&& node) const { return _pool->emplace(std::move(node)); }

    int clone_node(int t) const { return _pool->clone(t); }

    template <typename U>
    static T make_value(const U& value, int index) {
        if constexpr (requires(U x) { Monoid::make(x); }) {
            return Monoid::make(value);
        } else if constexpr (requires(U x, int i) { Monoid::make(x, i); }) {
            return Monoid::make(value, index);
        } else {
            return static_cast<T>(value);
        }
    }

    T compose_for_child(const T& inherited, const Node& node) const {
        if (!node.has_lazy) return inherited;
        return Monoid::op(inherited, node.val);
    }

    int build(int l, int r, const std::vector<T>& v) const {
        if (l == r) return 0;
        if (r - l == 1) return new_node(Node(v[l]));
        int m = (l + r) >> 1;
        return new_node(Node(build(l, m, v), build(m, r, v)));
    }

    int build(int l, int r, std::vector<T>& v) const {
        if (l == r) return 0;
        if (r - l == 1) return new_node(Node(std::move(v[l])));
        int m = (l + r) >> 1;
        return new_node(Node(build(l, m, v), build(m, r, v)));
    }

    template <typename U>
    int build_from_values(int l, int r, const std::vector<U>& v) const {
        if (l == r) return 0;
        if (r - l == 1) return new_node(Node(make_value(v[l], l)));
        int m = (l + r) >> 1;
        return new_node(Node(build_from_values(l, m, v), build_from_values(m, r, v)));
    }

    void all_apply_to_node(int t, const T& x, int l, int r) const {
        Node& node = (*_pool)[t];
        if (r - l == 1) {
            node.val = Monoid::op(x, node.val);
        } else {
            node.val = node.has_lazy ? Monoid::op(x, node.val) : x;
            node.has_lazy = true;
        }
    }

    int all_apply_clone(int t, const T& x, int l, int r, bool copy_on_write = false) const {
        int res = copy_on_write ? _pool->clone_if_shared(t) : clone_node(t);
        all_apply_to_node(res, x, l, r);
        return res;
    }

    void push(int t, int l, int r, bool copy_on_write = false) const {
        Node node = (*_pool)[t];
        if (!node.has_lazy || r - l == 1) return;
        int m = (l + r) >> 1;
        int left = all_apply_clone(node.left, node.val, l, m, copy_on_write);
        int right = all_apply_clone(node.right, node.val, m, r, copy_on_write);
        Node& target = (*_pool)[t];
        _pool->replace(target.left, left);
        _pool->replace(target.right, right);
        target.val = Monoid::id();
        target.has_lazy = false;
    }

    int set_node(int t, int l, int r, int p, T value, bool copy_on_write = false) const {
        t = copy_on_write ? _pool->clone_if_shared(t) : clone_node(t);
        if (r - l == 1) {
            Node& node = (*_pool)[t];
            node.val = std::move(value);
            node.has_lazy = false;
            return t;
        }
        push(t, l, r, copy_on_write);
        int m = (l + r) >> 1;
        if (p < m) {
            int child = set_node((*_pool)[t].left, l, m, p, std::move(value), copy_on_write);
            _pool->replace((*_pool)[t].left, child);
        } else {
            int child = set_node((*_pool)[t].right, m, r, p, std::move(value), copy_on_write);
            _pool->replace((*_pool)[t].right, child);
        }
        return t;
    }

    int apply_node(int t, int l, int r, int ql, int qr, const T& x, bool copy_on_write = false) const {
        if (qr <= l || r <= ql) return t;
        t = copy_on_write ? _pool->clone_if_shared(t) : clone_node(t);
        if (ql <= l && r <= qr) {
            all_apply_to_node(t, x, l, r);
            return t;
        }
        push(t, l, r, copy_on_write);
        int m = (l + r) >> 1;
        int left = apply_node((*_pool)[t].left, l, m, ql, qr, x, copy_on_write);
        int right = apply_node((*_pool)[t].right, m, r, ql, qr, x, copy_on_write);
        _pool->replace((*_pool)[t].left, left);
        _pool->replace((*_pool)[t].right, right);
        return t;
    }

    T get_node(int t, int l, int r, int p, const T& inherited) const {
        const Node& node = (*_pool)[t];
        if (r - l == 1) return Monoid::op(inherited, node.val);
        int m = (l + r) >> 1;
        if (p < m) return get_node(node.left, l, m, p, compose_for_child(inherited, node));
        return get_node(node.right, m, r, p, compose_for_child(inherited, node));
    }

    void collect_node(int t, int l, int r, int ql, int qr, const T& inherited, std::vector<T>& res) const {
        if (!t || qr <= l || r <= ql) return;
        const Node& node = (*_pool)[t];
        if (r - l == 1) {
            res.push_back(Monoid::op(inherited, node.val));
            return;
        }
        int m = (l + r) >> 1;
        T next = compose_for_child(inherited, node);
        collect_node(node.left, l, m, ql, qr, next, res);
        collect_node(node.right, m, r, ql, qr, next, res);
    }

   public:
    PersistentDualSegtree() : PersistentDualSegtree(0) {}

    explicit PersistentDualSegtree(int n) : _n(n), _root(0), _pool(std::make_shared<Pool>()) {
        assert(0 <= n);
        if (_n > 0) _root = build(0, _n, std::vector<T>(_n, Monoid::id()));
        _pool->retain(_root);
    }

    explicit PersistentDualSegtree(const std::vector<T>& v)
        : _n(int(v.size())), _root(0), _pool(std::make_shared<Pool>()) {
        _pool->reserve(v.size() * 2);
        if (_n > 0) _root = build(0, _n, v);
        _pool->retain(_root);
    }

    explicit PersistentDualSegtree(std::vector<T>&& v) : _n(int(v.size())), _root(0), _pool(std::make_shared<Pool>()) {
        _pool->reserve(v.size() * 2);
        if (_n > 0) _root = build(0, _n, v);
        _pool->retain(_root);
    }

    template <typename U>
        requires(!std::same_as<U, T>) &&
                (requires(U x) { Monoid::make(x); } || requires(U x, int i) { Monoid::make(x, i); } ||
                 std::convertible_to<U, T>)
    explicit PersistentDualSegtree(const std::vector<U>& v)
        : _n(int(v.size())), _root(0), _pool(std::make_shared<Pool>()) {
        _pool->reserve(v.size() * 2);
        if (_n > 0) _root = build_from_values(0, _n, v);
        _pool->retain(_root);
    }

    PersistentDualSegtree(const PersistentDualSegtree& other) : _n(other._n), _root(other._root), _pool(other._pool) {
        if (_pool) _pool->retain(_root);
    }
    PersistentDualSegtree(PersistentDualSegtree&& other) noexcept
        : _n(other._n), _root(other._root), _pool(std::move(other._pool)) {
        other._n = 0;
        other._root = 0;
    }
    PersistentDualSegtree& operator=(const PersistentDualSegtree& other) {
        if (this == &other) return *this;
        if (other._pool) other._pool->retain(other._root);
        if (_pool) _pool->release(_root);
        _n = other._n;
        _root = other._root;
        _pool = other._pool;
        return *this;
    }
    PersistentDualSegtree& operator=(PersistentDualSegtree&& other) noexcept {
        if (this == &other) return *this;
        if (_pool) _pool->release(_root);
        _n = other._n;
        _root = other._root;
        _pool = std::move(other._pool);
        other._n = 0;
        other._root = 0;
        return *this;
    }
    ~PersistentDualSegtree() {
        if (_pool) _pool->release(_root);
    }

    int size() const { return _n; }

    bool empty() const { return _n == 0; }

    void release() {
        if (_pool) _pool->release(_root);
        _pool = std::make_shared<Pool>();
        _root = 0;
        _n = 0;
    }

    std::size_t node_count() const { return _pool ? _pool->size() : 0; }

    PersistentDualSegtree set(int p, T x) const {
        assert(0 <= p && p < _n);
        return PersistentDualSegtree(_n, set_node(_root, 0, _n, p, std::move(x)), _pool);
    }

    void set_inplace(int p, T x) {
        assert(0 <= p && p < _n);
        int root = set_node(_root, 0, _n, p, std::move(x), true);
        _pool->replace(_root, root);
    }

    T get(int p) const {
        assert(0 <= p && p < _n);
        return get_node(_root, 0, _n, p, Monoid::id());
    }

    T operator[](int p) const { return get(p); }

    PersistentDualSegtree apply(int p, const T& x) const {
        assert(0 <= p && p < _n);
        return apply(p, p + 1, x);
    }

    PersistentDualSegtree apply(int l, int r, const T& x) const {
        assert(0 <= l && l <= r && r <= _n);
        if (l == r) return *this;
        return PersistentDualSegtree(_n, apply_node(_root, 0, _n, l, r, x), _pool);
    }

    void apply_inplace(int p, const T& x) {
        assert(0 <= p && p < _n);
        apply_inplace(p, p + 1, x);
    }

    void apply_inplace(int l, int r, const T& x) {
        assert(0 <= l && l <= r && r <= _n);
        if (l == r) return;
        int root = apply_node(_root, 0, _n, l, r, x, true);
        _pool->replace(_root, root);
    }

    std::vector<T> to_vector() const { return to_vector(0, _n); }

    std::vector<T> to_vector(int l, int r) const {
        assert(0 <= l && l <= r && r <= _n);
        std::vector<T> res;
        res.reserve(r - l);
        collect_node(_root, 0, _n, l, r, Monoid::id(), res);
        return res;
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "ds/segtree/persistent_dynamic_dual_segtree.hpp"



#line 9 "ds/segtree/persistent_dynamic_dual_segtree.hpp"
#include <numeric>
#include <type_traits>
#line 13 "ds/segtree/persistent_dynamic_dual_segtree.hpp"

#line 1 "ds/segtree/dynamic_segtree_common.hpp"



#line 11 "ds/segtree/dynamic_segtree_common.hpp"

namespace m1une {
namespace ds {
namespace detail {

template <std::integral Index>
using dynamic_size_type = std::make_unsigned_t<Index>;

template <std::integral Index>
constexpr dynamic_size_type<Index> dynamic_distance(Index left, Index right) {
    return static_cast<dynamic_size_type<Index>>(right) - static_cast<dynamic_size_type<Index>>(left);
}

template <class Monoid, class Size>
typename Monoid::value_type monoid_repeat(typename Monoid::value_type value, Size count) {
    typename Monoid::value_type result = Monoid::id();
    while (count != 0) {
        if (count & 1) result = Monoid::op(result, value);
        count >>= 1;
        if (count != 0) value = Monoid::op(value, value);
    }
    return result;
}

template <class ActedMonoid>
typename ActedMonoid::value_type dynamic_mapping(
    const typename ActedMonoid::operator_type& f,
    const typename ActedMonoid::value_type& value
) {
    using F = typename ActedMonoid::operator_type;
    using T = typename ActedMonoid::value_type;
    if constexpr (requires(F g, T x, long long ord) { ActedMonoid::mapping(g, x, ord); }) {
        return ActedMonoid::mapping(f, value, 0);
    } else {
        return ActedMonoid::mapping(f, value);
    }
}

template <class ActedMonoid, class Size>
typename ActedMonoid::operator_type dynamic_shift(
    const typename ActedMonoid::operator_type& f,
    Size offset
) {
    using F = typename ActedMonoid::operator_type;
    if constexpr (requires(F g, long long ord) { ActedMonoid::op_shift(g, ord); }) {
        assert(offset <= static_cast<Size>(std::numeric_limits<long long>::max()));
        return ActedMonoid::op_shift(f, static_cast<long long>(offset));
    } else {
        return f;
    }
}

template <class Monoid, std::integral Index>
class UniformMonoidDomain {
   public:
    using T = typename Monoid::value_type;
    using size_type = dynamic_size_type<Index>;

   private:
    struct Level {
        size_type small_length;
        T small_value;
        T large_value;
    };

    Index _left;
    Index _right;
    T _initial_value;
    std::vector<Level> _levels;

   public:
    UniformMonoidDomain(Index left, Index right, T initial_value)
        : _left(left), _right(right), _initial_value(std::move(initial_value)) {
        assert(left <= right);
        size_type n = size();
        constexpr int digits = std::numeric_limits<size_type>::digits;
        _levels.reserve(digits + 1);
        for (int depth = 0; depth <= digits; depth++) {
            size_type small = depth == digits ? 0 : n >> depth;
            size_type large = small;
            if (depth != 0) {
                bool has_remainder;
                if (depth == digits) {
                    has_remainder = n != 0;
                } else {
                    size_type mask = (size_type(1) << depth) - 1;
                    has_remainder = (n & mask) != 0;
                }
                if (has_remainder) large++;
            }
            _levels.push_back(Level{
                small,
                monoid_repeat<Monoid>(_initial_value, small),
                monoid_repeat<Monoid>(_initial_value, large),
            });
        }
    }

    Index left_bound() const {
        return _left;
    }

    Index right_bound() const {
        return _right;
    }

    size_type size() const {
        return dynamic_distance(_left, _right);
    }

    bool empty() const {
        return _left == _right;
    }

    const T& initial_value() const {
        return _initial_value;
    }

    const T& default_product(int depth, Index left, Index right) const {
        assert(0 <= depth && depth < int(_levels.size()));
        const Level& level = _levels[depth];
        size_type length = dynamic_distance(left, right);
        if (length == level.small_length) return level.small_value;
        assert(length == level.small_length + 1);
        return level.large_value;
    }
};

}  // namespace detail
}  // namespace ds
}  // namespace m1une


#line 17 "ds/segtree/persistent_dynamic_dual_segtree.hpp"

namespace m1une {
namespace ds {

// A persistent sparse dual segment tree over an integral half-open interval.
template <m1une::monoid::IsMonoid Monoid, std::integral Index = long long>
    requires(!std::same_as<std::remove_cv_t<Index>, bool>)
struct PersistentDynamicDualSegtree {
    using T = typename Monoid::value_type;
    using index_type = Index;
    using size_type = detail::dynamic_size_type<Index>;

   private:
    struct Node {
        T val;
        int left;
        int right;
        int references;
        bool has_lazy;

        Node() : val(Monoid::id()), left(0), right(0), references(0), has_lazy(false) {}
    };

    struct Config {
        Index left;
        Index right;
        T initial_value;

        Config(Index left_bound, Index right_bound, T value)
            : left(left_bound), right(right_bound), initial_value(std::move(value)) {
            assert(left <= right);
        }
    };

    std::shared_ptr<const Config> _config;
    using Pool = detail::PersistentNodePool<Node>;
    std::shared_ptr<Pool> _pool;
    int _root;

    PersistentDynamicDualSegtree(std::shared_ptr<const Config> config, std::shared_ptr<Pool> pool, int root)
        : _config(std::move(config)), _pool(std::move(pool)), _root(root) {
        _pool->retain(_root);
    }

    int new_node() const { return _pool->emplace(); }

    int clone_or_new(int t, bool copy_on_write = false) const {
        if (!t) return new_node();
        return copy_on_write ? _pool->clone_if_shared(t) : _pool->clone(t);
    }

    void all_apply_to_node(int t, Index left, Index right, const T& x) const {
        Node& node = (*_pool)[t];
        if (std::midpoint(left, right) == left) {
            T value = node.has_lazy ? node.val : _config->initial_value;
            node.val = Monoid::op(x, value);
            node.has_lazy = true;
        } else {
            node.val = node.has_lazy ? Monoid::op(x, node.val) : x;
            node.has_lazy = true;
        }
    }

    int all_apply_clone(int t, Index left, Index right, const T& x, bool copy_on_write = false) const {
        int result = clone_or_new(t, copy_on_write);
        all_apply_to_node(result, left, right, x);
        return result;
    }

    void push(int t, Index left, Index right, bool copy_on_write = false) const {
        if (!(*_pool)[t].has_lazy) return;
        Index middle = std::midpoint(left, right);
        if (middle == left) return;

        T lazy = (*_pool)[t].val;
        int left_child = all_apply_clone((*_pool)[t].left, left, middle, lazy, copy_on_write);
        int right_child = all_apply_clone((*_pool)[t].right, middle, right, lazy, copy_on_write);

        Node& node = (*_pool)[t];
        _pool->replace(node.left, left_child);
        _pool->replace(node.right, right_child);
        node.val = Monoid::id();
        node.has_lazy = false;
    }

    int set_node(int t, Index left, Index right, Index p, T x, bool copy_on_write = false) const {
        t = clone_or_new(t, copy_on_write);
        Index middle = std::midpoint(left, right);
        if (middle == left) {
            Node& node = (*_pool)[t];
            node.val = std::move(x);
            node.has_lazy = true;
            return t;
        }

        push(t, left, right, copy_on_write);
        if (p < middle) {
            int child = set_node((*_pool)[t].left, left, middle, p, std::move(x), copy_on_write);
            _pool->replace((*_pool)[t].left, child);
        } else {
            int child = set_node((*_pool)[t].right, middle, right, p, std::move(x), copy_on_write);
            _pool->replace((*_pool)[t].right, child);
        }
        return t;
    }

    int apply_node(int t, Index left, Index right, Index query_left, Index query_right, const T& x,
                   bool copy_on_write = false) const {
        if (query_right <= left || right <= query_left) return t;
        if (query_left <= left && right <= query_right) {
            return all_apply_clone(t, left, right, x, copy_on_write);
        }

        t = clone_or_new(t, copy_on_write);
        push(t, left, right, copy_on_write);
        Index middle = std::midpoint(left, right);
        int left_child = apply_node((*_pool)[t].left, left, middle, query_left, query_right, x, copy_on_write);
        int right_child = apply_node((*_pool)[t].right, middle, right, query_left, query_right, x, copy_on_write);
        _pool->replace((*_pool)[t].left, left_child);
        _pool->replace((*_pool)[t].right, right_child);
        return t;
    }

    T compose(const T& inherited, int t) const {
        if (!t || !(*_pool)[t].has_lazy) return inherited;
        return Monoid::op(inherited, (*_pool)[t].val);
    }

   public:
    PersistentDynamicDualSegtree() : PersistentDynamicDualSegtree(Index(0), Index(0), Monoid::id()) {}

    explicit PersistentDynamicDualSegtree(Index n) : PersistentDynamicDualSegtree(Index(0), n, Monoid::id()) {
        if constexpr (std::signed_integral<Index>) assert(Index(0) <= n);
    }

    PersistentDynamicDualSegtree(Index left, Index right) : PersistentDynamicDualSegtree(left, right, Monoid::id()) {}

    PersistentDynamicDualSegtree(Index left, Index right, T initial_value)
        : _config(std::make_shared<Config>(left, right, std::move(initial_value))),
          _pool(std::make_shared<Pool>()),
          _root(0) {}

    PersistentDynamicDualSegtree(const PersistentDynamicDualSegtree& other)
        : _config(other._config), _pool(other._pool), _root(other._root) {
        if (_pool) _pool->retain(_root);
    }
    PersistentDynamicDualSegtree(PersistentDynamicDualSegtree&& other) noexcept
        : _config(std::move(other._config)), _pool(std::move(other._pool)), _root(other._root) {
        other._root = 0;
    }
    PersistentDynamicDualSegtree& operator=(const PersistentDynamicDualSegtree& other) {
        if (this == &other) return *this;
        if (other._pool) other._pool->retain(other._root);
        if (_pool) _pool->release(_root);
        _config = other._config;
        _pool = other._pool;
        _root = other._root;
        return *this;
    }
    PersistentDynamicDualSegtree& operator=(PersistentDynamicDualSegtree&& other) noexcept {
        if (this == &other) return *this;
        if (_pool) _pool->release(_root);
        _config = std::move(other._config);
        _pool = std::move(other._pool);
        _root = other._root;
        other._root = 0;
        return *this;
    }
    ~PersistentDynamicDualSegtree() {
        if (_pool) _pool->release(_root);
    }

    size_type size() const { return detail::dynamic_distance(_config->left, _config->right); }

    bool empty() const { return _config->left == _config->right; }

    Index left_bound() const { return _config->left; }

    Index right_bound() const { return _config->right; }

    const T& initial_value() const { return _config->initial_value; }

    void reserve(std::size_t node_capacity) const {
        assert(node_capacity < std::numeric_limits<std::size_t>::max());
        _pool->reserve(node_capacity);
    }

    std::size_t node_count() const { return _pool->size(); }

    void release() {
        if (_pool) _pool->release(_root);
        _pool = std::make_shared<Pool>();
        _root = 0;
    }

    PersistentDynamicDualSegtree set(Index p, T x) const {
        assert(left_bound() <= p && p < right_bound());
        return PersistentDynamicDualSegtree(_config, _pool,
                                            set_node(_root, left_bound(), right_bound(), p, std::move(x)));
    }

    void set_inplace(Index p, T x) {
        assert(left_bound() <= p && p < right_bound());
        int root = set_node(_root, left_bound(), right_bound(), p, std::move(x), true);
        _pool->replace(_root, root);
    }

    T get(Index p) const {
        assert(left_bound() <= p && p < right_bound());
        int t = _root;
        Index left = left_bound();
        Index right = right_bound();
        T inherited = Monoid::id();

        while (t) {
            Index middle = std::midpoint(left, right);
            if (middle == left) {
                T value = (*_pool)[t].has_lazy ? (*_pool)[t].val : initial_value();
                return Monoid::op(inherited, value);
            }
            inherited = compose(inherited, t);
            if (p < middle) {
                t = (*_pool)[t].left;
                right = middle;
            } else {
                t = (*_pool)[t].right;
                left = middle;
            }
        }
        return Monoid::op(inherited, initial_value());
    }

    T operator[](Index p) const { return get(p); }

    PersistentDynamicDualSegtree apply(Index p, const T& x) const {
        assert(left_bound() <= p && p < right_bound());
        return apply(p, p + 1, x);
    }

    PersistentDynamicDualSegtree apply(Index left, Index right, const T& x) const {
        assert(left_bound() <= left && left <= right && right <= right_bound());
        if (left == right) return *this;
        return PersistentDynamicDualSegtree(_config, _pool,
                                            apply_node(_root, left_bound(), right_bound(), left, right, x));
    }

    void apply_inplace(Index p, const T& x) {
        assert(left_bound() <= p && p < right_bound());
        apply_inplace(p, p + 1, x);
    }

    void apply_inplace(Index left, Index right, const T& x) {
        assert(left_bound() <= left && left <= right && right <= right_bound());
        if (left == right) return;
        int root = apply_node(_root, left_bound(), right_bound(), left, right, x, true);
        _pool->replace(_root, root);
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "ds/segtree/persistent_dynamic_lazy_segtree.hpp"



#line 13 "ds/segtree/persistent_dynamic_lazy_segtree.hpp"

#line 17 "ds/segtree/persistent_dynamic_lazy_segtree.hpp"

namespace m1une {
namespace ds {

// A persistent sparse lazy segment tree over an integral half-open interval.
template <m1une::acted_monoid::IsActedMonoid ActedMonoid, std::integral Index = long long>
    requires(!std::same_as<std::remove_cv_t<Index>, bool>)
struct PersistentDynamicLazySegtree {
    using T = typename ActedMonoid::value_type;
    using F = typename ActedMonoid::operator_type;
    using index_type = Index;
    using size_type = detail::dynamic_size_type<Index>;

   private:
    struct Node {
        T val;
        F lazy;
        int left;
        int right;
        int references;
        bool has_lazy;

        Node()
            : val(ActedMonoid::id()), lazy(ActedMonoid::op_id()), left(0), right(0), references(0), has_lazy(false) {}
        explicit Node(T value)
            : val(std::move(value)), lazy(ActedMonoid::op_id()), left(0), right(0), references(0), has_lazy(false) {}
    };

    struct Config {
        detail::UniformMonoidDomain<ActedMonoid, Index> domain;

        Config(Index left, Index right, T initial_value) : domain(left, right, std::move(initial_value)) {}
    };

    std::shared_ptr<const Config> _config;
    using Pool = detail::PersistentNodePool<Node>;
    std::shared_ptr<Pool> _pool;
    int _root;

    PersistentDynamicLazySegtree(std::shared_ptr<const Config> config, std::shared_ptr<Pool> pool, int root)
        : _config(std::move(config)), _pool(std::move(pool)), _root(root) {
        _pool->retain(_root);
    }

    int new_node(Index left, Index right, int depth) const {
        return _pool->emplace(_config->domain.default_product(depth, left, right));
    }

    int clone_or_new(int t, Index left, Index right, int depth, bool copy_on_write = false) const {
        if (!t) return new_node(left, right, depth);
        return copy_on_write ? _pool->clone_if_shared(t) : _pool->clone(t);
    }

    const T& value(int t, Index left, Index right, int depth) const {
        if (t) return (*_pool)[t].val;
        return _config->domain.default_product(depth, left, right);
    }

    void all_apply_to_node(int t, Index left, Index right, const F& f) const {
        Node& node = (*_pool)[t];
        node.val = detail::dynamic_mapping<ActedMonoid>(f, node.val);
        if (std::midpoint(left, right) != left) {
            node.lazy = ActedMonoid::op_comp(f, node.lazy);
            node.has_lazy = true;
        }
    }

    int all_apply_clone(int t, Index left, Index right, int depth, const F& f,
                        bool copy_on_write = false) const {
        int result = clone_or_new(t, left, right, depth, copy_on_write);
        all_apply_to_node(result, left, right, f);
        return result;
    }

    void push(int t, Index left, Index right, int depth, bool copy_on_write = false) const {
        if (!(*_pool)[t].has_lazy) return;
        Index middle = std::midpoint(left, right);
        if (middle == left) return;

        F lazy = (*_pool)[t].lazy;
        int left_child = all_apply_clone((*_pool)[t].left, left, middle, depth + 1, lazy, copy_on_write);
        int right_child =
            all_apply_clone((*_pool)[t].right, middle, right, depth + 1,
                            detail::dynamic_shift<ActedMonoid>(lazy, detail::dynamic_distance(left, middle)),
                            copy_on_write);

        Node& node = (*_pool)[t];
        _pool->replace(node.left, left_child);
        _pool->replace(node.right, right_child);
        node.lazy = ActedMonoid::op_id();
        node.has_lazy = false;
    }

    void update(int t, Index left, Index right, int depth) const {
        Index middle = std::midpoint(left, right);
        Node& node = (*_pool)[t];
        node.val =
            ActedMonoid::op(value(node.left, left, middle, depth + 1), value(node.right, middle, right, depth + 1));
    }

    int set_node(int t, Index left, Index right, int depth, Index p, T x,
                 bool copy_on_write = false) const {
        t = clone_or_new(t, left, right, depth, copy_on_write);
        Index middle = std::midpoint(left, right);
        if (middle == left) {
            Node& node = (*_pool)[t];
            node.val = std::move(x);
            node.lazy = ActedMonoid::op_id();
            node.has_lazy = false;
            return t;
        }

        push(t, left, right, depth, copy_on_write);
        if (p < middle) {
            int child = set_node((*_pool)[t].left, left, middle, depth + 1, p, std::move(x), copy_on_write);
            _pool->replace((*_pool)[t].left, child);
        } else {
            int child = set_node((*_pool)[t].right, middle, right, depth + 1, p, std::move(x), copy_on_write);
            _pool->replace((*_pool)[t].right, child);
        }
        update(t, left, right, depth);
        return t;
    }

    int apply_node(int t, Index left, Index right, int depth, Index query_left, Index query_right, const F& f,
                   bool copy_on_write = false) const {
        if (query_right <= left || right <= query_left) return t;
        if (query_left <= left && right <= query_right) {
            return all_apply_clone(t, left, right, depth,
                                   detail::dynamic_shift<ActedMonoid>(f, detail::dynamic_distance(query_left, left)),
                                   copy_on_write);
        }

        t = clone_or_new(t, left, right, depth, copy_on_write);
        push(t, left, right, depth, copy_on_write);
        Index middle = std::midpoint(left, right);
        int left_child = apply_node((*_pool)[t].left, left, middle, depth + 1, query_left, query_right, f,
                                    copy_on_write);
        int right_child = apply_node((*_pool)[t].right, middle, right, depth + 1, query_left, query_right, f,
                                     copy_on_write);
        _pool->replace((*_pool)[t].left, left_child);
        _pool->replace((*_pool)[t].right, right_child);
        update(t, left, right, depth);
        return t;
    }

    F compose_for_child(const F& inherited, int t, size_type offset) const {
        F shifted = detail::dynamic_shift<ActedMonoid>(inherited, offset);
        if (!t || !(*_pool)[t].has_lazy) return shifted;
        return ActedMonoid::op_comp(shifted, detail::dynamic_shift<ActedMonoid>((*_pool)[t].lazy, offset));
    }

    T prod_node(int t, Index left, Index right, int depth, Index query_left, Index query_right,
                const F& inherited) const {
        if (query_right <= left || right <= query_left) return ActedMonoid::id();
        if (query_left <= left && right <= query_right) {
            return detail::dynamic_mapping<ActedMonoid>(inherited, value(t, left, right, depth));
        }
        Index middle = std::midpoint(left, right);
        return ActedMonoid::op(prod_node(t ? (*_pool)[t].left : 0, left, middle, depth + 1, query_left, query_right,
                                         compose_for_child(inherited, t, 0)),
                               prod_node(t ? (*_pool)[t].right : 0, middle, right, depth + 1, query_left, query_right,
                                         compose_for_child(inherited, t, detail::dynamic_distance(left, middle))));
    }

    template <class G>
    Index max_right_node(int t, Index left, Index right, int depth, Index query_left, T& product, const F& inherited,
                         G& predicate) const {
        if (right <= query_left) return right;
        if (query_left <= left) {
            T next =
                ActedMonoid::op(product, detail::dynamic_mapping<ActedMonoid>(inherited, value(t, left, right, depth)));
            if (predicate(next)) {
                product = std::move(next);
                return right;
            }
            Index middle = std::midpoint(left, right);
            if (middle == left) return left;
        }
        Index middle = std::midpoint(left, right);
        Index result = max_right_node(t ? (*_pool)[t].left : 0, left, middle, depth + 1, query_left, product,
                                      compose_for_child(inherited, t, 0), predicate);
        if (result < middle) return result;
        return max_right_node(t ? (*_pool)[t].right : 0, middle, right, depth + 1, query_left, product,
                              compose_for_child(inherited, t, detail::dynamic_distance(left, middle)), predicate);
    }

    template <class G>
    Index min_left_node(int t, Index left, Index right, int depth, Index query_right, T& product, const F& inherited,
                        G& predicate) const {
        if (query_right <= left) return left;
        if (right <= query_right) {
            T next =
                ActedMonoid::op(detail::dynamic_mapping<ActedMonoid>(inherited, value(t, left, right, depth)), product);
            if (predicate(next)) {
                product = std::move(next);
                return left;
            }
            Index middle = std::midpoint(left, right);
            if (middle == left) return right;
        }
        Index middle = std::midpoint(left, right);
        Index result =
            min_left_node(t ? (*_pool)[t].right : 0, middle, right, depth + 1, query_right, product,
                          compose_for_child(inherited, t, detail::dynamic_distance(left, middle)), predicate);
        if (middle < result) return result;
        return min_left_node(t ? (*_pool)[t].left : 0, left, middle, depth + 1, query_right, product,
                             compose_for_child(inherited, t, 0), predicate);
    }

   public:
    PersistentDynamicLazySegtree() : PersistentDynamicLazySegtree(Index(0), Index(0), ActedMonoid::id()) {}

    explicit PersistentDynamicLazySegtree(Index n) : PersistentDynamicLazySegtree(Index(0), n, ActedMonoid::id()) {
        if constexpr (std::signed_integral<Index>) assert(Index(0) <= n);
    }

    PersistentDynamicLazySegtree(Index left, Index right)
        : PersistentDynamicLazySegtree(left, right, ActedMonoid::id()) {}

    PersistentDynamicLazySegtree(Index left, Index right, T initial_value)
        : _config(std::make_shared<Config>(left, right, std::move(initial_value))),
          _pool(std::make_shared<Pool>()),
          _root(0) {}

    PersistentDynamicLazySegtree(const PersistentDynamicLazySegtree& other)
        : _config(other._config), _pool(other._pool), _root(other._root) {
        if (_pool) _pool->retain(_root);
    }
    PersistentDynamicLazySegtree(PersistentDynamicLazySegtree&& other) noexcept
        : _config(std::move(other._config)), _pool(std::move(other._pool)), _root(other._root) {
        other._root = 0;
    }
    PersistentDynamicLazySegtree& operator=(const PersistentDynamicLazySegtree& other) {
        if (this == &other) return *this;
        if (other._pool) other._pool->retain(other._root);
        if (_pool) _pool->release(_root);
        _config = other._config;
        _pool = other._pool;
        _root = other._root;
        return *this;
    }
    PersistentDynamicLazySegtree& operator=(PersistentDynamicLazySegtree&& other) noexcept {
        if (this == &other) return *this;
        if (_pool) _pool->release(_root);
        _config = std::move(other._config);
        _pool = std::move(other._pool);
        _root = other._root;
        other._root = 0;
        return *this;
    }
    ~PersistentDynamicLazySegtree() {
        if (_pool) _pool->release(_root);
    }

    size_type size() const { return _config->domain.size(); }

    bool empty() const { return _config->domain.empty(); }

    Index left_bound() const { return _config->domain.left_bound(); }

    Index right_bound() const { return _config->domain.right_bound(); }

    const T& initial_value() const { return _config->domain.initial_value(); }

    void reserve(std::size_t node_capacity) const {
        assert(node_capacity < std::numeric_limits<std::size_t>::max());
        _pool->reserve(node_capacity);
    }

    std::size_t node_count() const { return _pool->size(); }

    void release() {
        if (_pool) _pool->release(_root);
        _pool = std::make_shared<Pool>();
        _root = 0;
    }

    PersistentDynamicLazySegtree set(Index p, T x) const {
        assert(left_bound() <= p && p < right_bound());
        return PersistentDynamicLazySegtree(_config, _pool,
                                            set_node(_root, left_bound(), right_bound(), 0, p, std::move(x)));
    }

    void set_inplace(Index p, T x) {
        assert(left_bound() <= p && p < right_bound());
        int root = set_node(_root, left_bound(), right_bound(), 0, p, std::move(x), true);
        _pool->replace(_root, root);
    }

    T get(Index p) const {
        assert(left_bound() <= p && p < right_bound());
        return prod(p, p + 1);
    }

    T operator[](Index p) const { return get(p); }

    T prod(Index left, Index right) const {
        assert(left_bound() <= left && left <= right && right <= right_bound());
        if (left == right) return ActedMonoid::id();
        return prod_node(_root, left_bound(), right_bound(), 0, left, right, ActedMonoid::op_id());
    }

    T all_prod() const { return value(_root, left_bound(), right_bound(), 0); }

    PersistentDynamicLazySegtree apply(Index p, const F& f) const {
        assert(left_bound() <= p && p < right_bound());
        return apply(p, p + 1, f);
    }

    PersistentDynamicLazySegtree apply(Index left, Index right, const F& f) const {
        assert(left_bound() <= left && left <= right && right <= right_bound());
        if (left == right) return *this;
        return PersistentDynamicLazySegtree(_config, _pool,
                                            apply_node(_root, left_bound(), right_bound(), 0, left, right, f));
    }

    void apply_inplace(Index p, const F& f) {
        assert(left_bound() <= p && p < right_bound());
        apply_inplace(p, p + 1, f);
    }

    void apply_inplace(Index left, Index right, const F& f) {
        assert(left_bound() <= left && left <= right && right <= right_bound());
        if (left == right) return;
        int root = apply_node(_root, left_bound(), right_bound(), 0, left, right, f, true);
        _pool->replace(_root, root);
    }

    template <class G>
    Index max_right(Index left, G predicate) const {
        assert(left_bound() <= left && left <= right_bound());
        assert(predicate(ActedMonoid::id()));
        if (left == right_bound()) return right_bound();
        T product = ActedMonoid::id();
        return max_right_node(_root, left_bound(), right_bound(), 0, left, product, ActedMonoid::op_id(), predicate);
    }

    template <class G>
    Index min_left(Index right, G predicate) const {
        assert(left_bound() <= right && right <= right_bound());
        assert(predicate(ActedMonoid::id()));
        if (right == left_bound()) return left_bound();
        T product = ActedMonoid::id();
        return min_left_node(_root, left_bound(), right_bound(), 0, right, product, ActedMonoid::op_id(), predicate);
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "ds/segtree/persistent_dynamic_segtree.hpp"



#line 13 "ds/segtree/persistent_dynamic_segtree.hpp"

#line 17 "ds/segtree/persistent_dynamic_segtree.hpp"

namespace m1une {
namespace ds {

// A persistent sparse segment tree over an integral half-open interval.
template <m1une::monoid::IsMonoid Monoid, std::integral Index = long long>
    requires(!std::same_as<std::remove_cv_t<Index>, bool>)
struct PersistentDynamicSegtree {
    using T = typename Monoid::value_type;
    using index_type = Index;
    using size_type = detail::dynamic_size_type<Index>;

   private:
    struct Node {
        T val;
        int left;
        int right;
        int references;

        Node() : val(Monoid::id()), left(0), right(0), references(0) {}
        explicit Node(T value) : val(std::move(value)), left(0), right(0), references(0) {}
    };

    struct Config {
        detail::UniformMonoidDomain<Monoid, Index> domain;

        Config(Index left, Index right, T initial_value) : domain(left, right, std::move(initial_value)) {}
    };

    std::shared_ptr<const Config> _config;
    using Pool = detail::PersistentNodePool<Node>;
    std::shared_ptr<Pool> _pool;
    int _root;

    PersistentDynamicSegtree(std::shared_ptr<const Config> config, std::shared_ptr<Pool> pool, int root)
        : _config(std::move(config)), _pool(std::move(pool)), _root(root) {
        _pool->retain(_root);
    }

    int new_node(T value) const { return _pool->emplace(std::move(value)); }

    const T& value(int t, Index left, Index right, int depth) const {
        if (t) return (*_pool)[t].val;
        return _config->domain.default_product(depth, left, right);
    }

    int set_node(int t, Index left, Index right, int depth, Index p, T x, bool copy_on_write = false) const {
        Index middle = std::midpoint(left, right);
        if (copy_on_write) {
            int result = t ? _pool->clone_if_shared(t) : new_node(value(0, left, right, depth));
            if (middle == left) {
                (*_pool)[result].val = std::move(x);
                return result;
            }
            int child;
            if (p < middle) {
                child = set_node((*_pool)[result].left, left, middle, depth + 1, p, std::move(x), true);
                _pool->replace((*_pool)[result].left, child);
            } else {
                child = set_node((*_pool)[result].right, middle, right, depth + 1, p, std::move(x), true);
                _pool->replace((*_pool)[result].right, child);
            }
            Node& node = (*_pool)[result];
            node.val = Monoid::op(value(node.left, left, middle, depth + 1),
                                  value(node.right, middle, right, depth + 1));
            return result;
        }
        if (middle == left) return new_node(std::move(x));

        int left_child = t ? (*_pool)[t].left : 0;
        int right_child = t ? (*_pool)[t].right : 0;
        if (p < middle) {
            left_child = set_node(left_child, left, middle, depth + 1, p, std::move(x));
        } else {
            right_child = set_node(right_child, middle, right, depth + 1, p, std::move(x));
        }

        int result = new_node(
            Monoid::op(value(left_child, left, middle, depth + 1), value(right_child, middle, right, depth + 1)));
        _pool->replace((*_pool)[result].left, left_child);
        _pool->replace((*_pool)[result].right, right_child);
        return result;
    }

    T prod_node(int t, Index left, Index right, int depth, Index query_left, Index query_right) const {
        if (query_right <= left || right <= query_left) return Monoid::id();
        if (query_left <= left && right <= query_right) {
            return value(t, left, right, depth);
        }
        Index middle = std::midpoint(left, right);
        return Monoid::op(prod_node(t ? (*_pool)[t].left : 0, left, middle, depth + 1, query_left, query_right),
                          prod_node(t ? (*_pool)[t].right : 0, middle, right, depth + 1, query_left, query_right));
    }

    template <class F>
    Index max_right_node(int t, Index left, Index right, int depth, Index query_left, T& product, F& predicate) const {
        if (right <= query_left) return right;
        if (query_left <= left) {
            T next = Monoid::op(product, value(t, left, right, depth));
            if (predicate(next)) {
                product = std::move(next);
                return right;
            }
            Index middle = std::midpoint(left, right);
            if (middle == left) return left;
        }

        Index middle = std::midpoint(left, right);
        Index result =
            max_right_node(t ? (*_pool)[t].left : 0, left, middle, depth + 1, query_left, product, predicate);
        if (result < middle) return result;
        return max_right_node(t ? (*_pool)[t].right : 0, middle, right, depth + 1, query_left, product, predicate);
    }

    template <class F>
    Index min_left_node(int t, Index left, Index right, int depth, Index query_right, T& product, F& predicate) const {
        if (query_right <= left) return left;
        if (right <= query_right) {
            T next = Monoid::op(value(t, left, right, depth), product);
            if (predicate(next)) {
                product = std::move(next);
                return left;
            }
            Index middle = std::midpoint(left, right);
            if (middle == left) return right;
        }

        Index middle = std::midpoint(left, right);
        Index result =
            min_left_node(t ? (*_pool)[t].right : 0, middle, right, depth + 1, query_right, product, predicate);
        if (middle < result) return result;
        return min_left_node(t ? (*_pool)[t].left : 0, left, middle, depth + 1, query_right, product, predicate);
    }

   public:
    PersistentDynamicSegtree() : PersistentDynamicSegtree(Index(0), Index(0), Monoid::id()) {}

    explicit PersistentDynamicSegtree(Index n) : PersistentDynamicSegtree(Index(0), n, Monoid::id()) {
        if constexpr (std::signed_integral<Index>) assert(Index(0) <= n);
    }

    PersistentDynamicSegtree(Index left, Index right) : PersistentDynamicSegtree(left, right, Monoid::id()) {}

    PersistentDynamicSegtree(Index left, Index right, T initial_value)
        : _config(std::make_shared<Config>(left, right, std::move(initial_value))),
          _pool(std::make_shared<Pool>()),
          _root(0) {}

    PersistentDynamicSegtree(const PersistentDynamicSegtree& other)
        : _config(other._config), _pool(other._pool), _root(other._root) {
        if (_pool) _pool->retain(_root);
    }
    PersistentDynamicSegtree(PersistentDynamicSegtree&& other) noexcept
        : _config(std::move(other._config)), _pool(std::move(other._pool)), _root(other._root) {
        other._root = 0;
    }
    PersistentDynamicSegtree& operator=(const PersistentDynamicSegtree& other) {
        if (this == &other) return *this;
        if (other._pool) other._pool->retain(other._root);
        if (_pool) _pool->release(_root);
        _config = other._config;
        _pool = other._pool;
        _root = other._root;
        return *this;
    }
    PersistentDynamicSegtree& operator=(PersistentDynamicSegtree&& other) noexcept {
        if (this == &other) return *this;
        if (_pool) _pool->release(_root);
        _config = std::move(other._config);
        _pool = std::move(other._pool);
        _root = other._root;
        other._root = 0;
        return *this;
    }
    ~PersistentDynamicSegtree() {
        if (_pool) _pool->release(_root);
    }

    size_type size() const { return _config->domain.size(); }

    bool empty() const { return _config->domain.empty(); }

    Index left_bound() const { return _config->domain.left_bound(); }

    Index right_bound() const { return _config->domain.right_bound(); }

    const T& initial_value() const { return _config->domain.initial_value(); }

    void reserve(std::size_t node_capacity) const {
        assert(node_capacity < std::numeric_limits<std::size_t>::max());
        _pool->reserve(node_capacity);
    }

    std::size_t node_count() const { return _pool->size(); }

    void release() {
        if (_pool) _pool->release(_root);
        _pool = std::make_shared<Pool>();
        _root = 0;
    }

    PersistentDynamicSegtree set(Index p, T x) const {
        assert(left_bound() <= p && p < right_bound());
        return PersistentDynamicSegtree(_config, _pool,
                                        set_node(_root, left_bound(), right_bound(), 0, p, std::move(x)));
    }

    void set_inplace(Index p, T x) {
        assert(left_bound() <= p && p < right_bound());
        int root = set_node(_root, left_bound(), right_bound(), 0, p, std::move(x), true);
        _pool->replace(_root, root);
    }

    T get(Index p) const {
        assert(left_bound() <= p && p < right_bound());
        int t = _root;
        Index left = left_bound();
        Index right = right_bound();

        while (t) {
            Index middle = std::midpoint(left, right);
            if (middle == left) return (*_pool)[t].val;
            if (p < middle) {
                t = (*_pool)[t].left;
                right = middle;
            } else {
                t = (*_pool)[t].right;
                left = middle;
            }
        }
        return initial_value();
    }

    T operator[](Index p) const { return get(p); }

    T prod(Index left, Index right) const {
        assert(left_bound() <= left && left <= right && right <= right_bound());
        if (left == right) return Monoid::id();
        return prod_node(_root, left_bound(), right_bound(), 0, left, right);
    }

    T all_prod() const { return value(_root, left_bound(), right_bound(), 0); }

    template <class F>
    Index max_right(Index left, F predicate) const {
        assert(left_bound() <= left && left <= right_bound());
        assert(predicate(Monoid::id()));
        if (left == right_bound()) return right_bound();
        T product = Monoid::id();
        return max_right_node(_root, left_bound(), right_bound(), 0, left, product, predicate);
    }

    template <class F>
    Index min_left(Index right, F predicate) const {
        assert(left_bound() <= right && right <= right_bound());
        assert(predicate(Monoid::id()));
        if (right == left_bound()) return left_bound();
        T product = Monoid::id();
        return min_left_node(_root, left_bound(), right_bound(), 0, right, product, predicate);
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "ds/segtree/persistent_lazy_segtree.hpp"



#line 9 "ds/segtree/persistent_lazy_segtree.hpp"

#line 12 "ds/segtree/persistent_lazy_segtree.hpp"

namespace m1une {
namespace ds {

template <m1une::acted_monoid::IsActedMonoid ActedMonoid>
struct PersistentLazySegtree {
    using T = typename ActedMonoid::value_type;
    using F = typename ActedMonoid::operator_type;

   private:
    struct Node {
        T val;
        F lazy;
        int left, right;
        int references;
        bool has_lazy;

        Node()
            : val(ActedMonoid::id()), lazy(ActedMonoid::op_id()), left(0), right(0), references(0), has_lazy(false) {}
        explicit Node(T value)
            : val(std::move(value)), lazy(ActedMonoid::op_id()), left(0), right(0), references(0), has_lazy(false) {}
        Node(T value, int left_child, int right_child)
            : val(std::move(value)),
              lazy(ActedMonoid::op_id()),
              left(left_child),
              right(right_child),
              references(0),
              has_lazy(false) {}
    };

    using Pool = detail::PersistentNodePool<Node>;

    int _n;
    int _root;
    std::shared_ptr<Pool> _pool;

    explicit PersistentLazySegtree(int n, int root, std::shared_ptr<Pool> pool)
        : _n(n), _root(root), _pool(std::move(pool)) {
        _pool->retain(_root);
    }

    int new_node(const Node& node) const { return _pool->emplace(node); }

    int new_node(Node&& node) const { return _pool->emplace(std::move(node)); }

    int clone_node(int t) const { return _pool->clone(t); }

    template <typename U>
    static T make_value(const U& value, int index) {
        if constexpr (requires(U x) { ActedMonoid::make(x); }) {
            return ActedMonoid::make(value);
        } else if constexpr (requires(U x, int i) { ActedMonoid::make(x, i); }) {
            return ActedMonoid::make(value, index);
        } else {
            return static_cast<T>(value);
        }
    }

    static T mapping_at(const F& f, const T& value, long long ord) {
        if constexpr (requires(F g, T x, long long i) { ActedMonoid::mapping(g, x, i); }) {
            return ActedMonoid::mapping(f, value, ord);
        } else {
            return ActedMonoid::mapping(f, value);
        }
    }

    static F shift_operator(const F& f, long long ord) {
        if constexpr (requires(F g, long long i) { ActedMonoid::op_shift(g, i); }) {
            return ActedMonoid::op_shift(f, ord);
        } else {
            return f;
        }
    }

    F compose_for_child(const F& inherited, const Node& node, long long ord) const {
        F shifted = shift_operator(inherited, ord);
        if (!node.has_lazy) return shifted;
        return ActedMonoid::op_comp(shifted, shift_operator(node.lazy, ord));
    }

    int build(int l, int r, const std::vector<T>& v) const {
        if (l == r) return 0;
        if (r - l == 1) return new_node(Node(v[l]));
        int m = (l + r) >> 1;
        int left = build(l, m, v);
        int right = build(m, r, v);
        return new_node(Node(ActedMonoid::op((*_pool)[left].val, (*_pool)[right].val), left, right));
    }

    int build(int l, int r, std::vector<T>& v) const {
        if (l == r) return 0;
        if (r - l == 1) return new_node(Node(std::move(v[l])));
        int m = (l + r) >> 1;
        int left = build(l, m, v);
        int right = build(m, r, v);
        return new_node(Node(ActedMonoid::op((*_pool)[left].val, (*_pool)[right].val), left, right));
    }

    template <typename U>
    int build_from_values(int l, int r, const std::vector<U>& v) const {
        if (l == r) return 0;
        if (r - l == 1) return new_node(Node(make_value(v[l], l)));
        int m = (l + r) >> 1;
        int left = build_from_values(l, m, v);
        int right = build_from_values(m, r, v);
        return new_node(Node(ActedMonoid::op((*_pool)[left].val, (*_pool)[right].val), left, right));
    }

    void all_apply_to_node(int t, const F& f) const {
        Node& node = (*_pool)[t];
        node.val = mapping_at(f, node.val, 0);
        node.lazy = ActedMonoid::op_comp(f, node.lazy);
        node.has_lazy = true;
    }

    int all_apply_clone(int t, const F& f, bool copy_on_write = false) const {
        int res = copy_on_write ? _pool->clone_if_shared(t) : clone_node(t);
        all_apply_to_node(res, f);
        return res;
    }

    void push(int t, int l, int r, bool copy_on_write = false) const {
        if (!(*_pool)[t].has_lazy) return;
        F lazy = (*_pool)[t].lazy;
        int left = (*_pool)[t].left;
        int right = (*_pool)[t].right;
        int m = (l + r) >> 1;
        left = all_apply_clone(left, lazy, copy_on_write);
        right = all_apply_clone(right, shift_operator(lazy, m - l), copy_on_write);
        Node& node = (*_pool)[t];
        _pool->replace(node.left, left);
        _pool->replace(node.right, right);
        node.lazy = ActedMonoid::op_id();
        node.has_lazy = false;
    }

    void update(int t) const {
        Node& node = (*_pool)[t];
        node.val = ActedMonoid::op((*_pool)[node.left].val, (*_pool)[node.right].val);
    }

    int set_node(int t, int l, int r, int p, T value, bool copy_on_write = false) const {
        t = copy_on_write ? _pool->clone_if_shared(t) : clone_node(t);
        if (r - l == 1) {
            Node& node = (*_pool)[t];
            node.val = std::move(value);
            node.lazy = ActedMonoid::op_id();
            node.has_lazy = false;
            return t;
        }
        push(t, l, r, copy_on_write);
        int m = (l + r) >> 1;
        if (p < m) {
            int child = set_node((*_pool)[t].left, l, m, p, std::move(value), copy_on_write);
            _pool->replace((*_pool)[t].left, child);
        } else {
            int child = set_node((*_pool)[t].right, m, r, p, std::move(value), copy_on_write);
            _pool->replace((*_pool)[t].right, child);
        }
        update(t);
        return t;
    }

    int apply_node(int t, int l, int r, int ql, int qr, const F& f, bool copy_on_write = false) const {
        if (qr <= l || r <= ql) return t;
        t = copy_on_write ? _pool->clone_if_shared(t) : clone_node(t);
        if (ql <= l && r <= qr) {
            all_apply_to_node(t, shift_operator(f, l - ql));
            return t;
        }
        push(t, l, r, copy_on_write);
        int m = (l + r) >> 1;
        int left = apply_node((*_pool)[t].left, l, m, ql, qr, f, copy_on_write);
        int right = apply_node((*_pool)[t].right, m, r, ql, qr, f, copy_on_write);
        _pool->replace((*_pool)[t].left, left);
        _pool->replace((*_pool)[t].right, right);
        update(t);
        return t;
    }

    int copy_range_node(int target, int source, int l, int r, int ql, int qr) const {
        if (qr <= l || r <= ql) return target;
        if (ql <= l && r <= qr) return source;

        target = clone_node(target);
        source = clone_node(source);
        _pool->retain(source);
        push(target, l, r);
        push(source, l, r);

        int m = (l + r) >> 1;
        int left = copy_range_node((*_pool)[target].left, (*_pool)[source].left, l, m, ql, qr);
        int right = copy_range_node((*_pool)[target].right, (*_pool)[source].right, m, r, ql, qr);
        _pool->replace((*_pool)[target].left, left);
        _pool->replace((*_pool)[target].right, right);
        update(target);
        _pool->release(source);
        return target;
    }

    T prod_node(int t, int l, int r, int ql, int qr, const F& inherited) const {
        if (!t || qr <= l || r <= ql) return ActedMonoid::id();
        const Node& node = (*_pool)[t];
        if (ql <= l && r <= qr) return mapping_at(inherited, node.val, 0);
        int m = (l + r) >> 1;
        return ActedMonoid::op(prod_node(node.left, l, m, ql, qr, compose_for_child(inherited, node, 0)),
                               prod_node(node.right, m, r, ql, qr, compose_for_child(inherited, node, m - l)));
    }

    void collect_node(int t, int l, int r, int ql, int qr, const F& inherited, std::vector<T>& res) const {
        if (!t || qr <= l || r <= ql) return;
        const Node& node = (*_pool)[t];
        if (r - l == 1) {
            res.push_back(mapping_at(inherited, node.val, 0));
            return;
        }
        int m = (l + r) >> 1;
        collect_node(node.left, l, m, ql, qr, compose_for_child(inherited, node, 0), res);
        collect_node(node.right, m, r, ql, qr, compose_for_child(inherited, node, m - l), res);
    }

    template <class G>
    int max_right_node(int t, int l, int r, int ql, T& sm, const F& inherited, G& g) const {
        if (r <= ql) return r;
        const Node& node = (*_pool)[t];
        if (ql <= l) {
            T nxt = ActedMonoid::op(sm, mapping_at(inherited, node.val, 0));
            if (g(nxt)) {
                sm = std::move(nxt);
                return r;
            }
            if (r - l == 1) return l;
        }
        int m = (l + r) >> 1;
        int res = max_right_node(node.left, l, m, ql, sm, compose_for_child(inherited, node, 0), g);
        if (res < m) return res;
        return max_right_node(node.right, m, r, ql, sm, compose_for_child(inherited, node, m - l), g);
    }

    template <class G>
    int min_left_node(int t, int l, int r, int qr, T& sm, const F& inherited, G& g) const {
        if (qr <= l) return l;
        const Node& node = (*_pool)[t];
        if (r <= qr) {
            T nxt = ActedMonoid::op(mapping_at(inherited, node.val, 0), sm);
            if (g(nxt)) {
                sm = std::move(nxt);
                return l;
            }
            if (r - l == 1) return r;
        }
        int m = (l + r) >> 1;
        int res = min_left_node(node.right, m, r, qr, sm, compose_for_child(inherited, node, m - l), g);
        if (m < res) return res;
        return min_left_node(node.left, l, m, qr, sm, compose_for_child(inherited, node, 0), g);
    }

   public:
    PersistentLazySegtree() : PersistentLazySegtree(0) {}

    explicit PersistentLazySegtree(int n) : _n(n), _root(0), _pool(std::make_shared<Pool>()) {
        assert(0 <= n);
        if (_n > 0) _root = build(0, _n, std::vector<T>(_n, ActedMonoid::id()));
        _pool->retain(_root);
    }

    explicit PersistentLazySegtree(const std::vector<T>& v)
        : _n(int(v.size())), _root(0), _pool(std::make_shared<Pool>()) {
        _pool->reserve(v.size() * 2);
        if (_n > 0) _root = build(0, _n, v);
        _pool->retain(_root);
    }

    explicit PersistentLazySegtree(std::vector<T>&& v) : _n(int(v.size())), _root(0), _pool(std::make_shared<Pool>()) {
        _pool->reserve(v.size() * 2);
        if (_n > 0) _root = build(0, _n, v);
        _pool->retain(_root);
    }

    template <typename U>
        requires(!std::same_as<U, T>) &&
                (requires(U x) { ActedMonoid::make(x); } || requires(U x, int i) { ActedMonoid::make(x, i); } ||
                 std::convertible_to<U, T>)
    explicit PersistentLazySegtree(const std::vector<U>& v)
        : _n(int(v.size())), _root(0), _pool(std::make_shared<Pool>()) {
        _pool->reserve(v.size() * 2);
        if (_n > 0) _root = build_from_values(0, _n, v);
        _pool->retain(_root);
    }

    PersistentLazySegtree(const PersistentLazySegtree& other) : _n(other._n), _root(other._root), _pool(other._pool) {
        if (_pool) _pool->retain(_root);
    }

    PersistentLazySegtree(PersistentLazySegtree&& other) noexcept
        : _n(other._n), _root(other._root), _pool(std::move(other._pool)) {
        other._n = 0;
        other._root = 0;
    }

    PersistentLazySegtree& operator=(const PersistentLazySegtree& other) {
        if (this == &other) return *this;
        if (other._pool) other._pool->retain(other._root);
        if (_pool) _pool->release(_root);
        _n = other._n;
        _root = other._root;
        _pool = other._pool;
        return *this;
    }

    PersistentLazySegtree& operator=(PersistentLazySegtree&& other) noexcept {
        if (this == &other) return *this;
        if (_pool) _pool->release(_root);
        _n = other._n;
        _root = other._root;
        _pool = std::move(other._pool);
        other._n = 0;
        other._root = 0;
        return *this;
    }

    ~PersistentLazySegtree() {
        if (_pool) _pool->release(_root);
    }

    int size() const { return _n; }

    bool empty() const { return _n == 0; }

    void release() {
        if (_pool) _pool->release(_root);
        _pool = std::make_shared<Pool>();
        _root = 0;
        _n = 0;
    }

    std::size_t node_count() const { return _pool ? _pool->size() : 0; }

    PersistentLazySegtree set(int p, T x) const {
        assert(0 <= p && p < _n);
        return PersistentLazySegtree(_n, set_node(_root, 0, _n, p, std::move(x)), _pool);
    }

    void set_inplace(int p, T x) {
        assert(0 <= p && p < _n);
        int root = set_node(_root, 0, _n, p, std::move(x), true);
        _pool->replace(_root, root);
    }

    T get(int p) const {
        assert(0 <= p && p < _n);
        return prod(p, p + 1);
    }

    T operator[](int p) const { return get(p); }

    T prod(int l, int r) const {
        assert(0 <= l && l <= r && r <= _n);
        if (l == r) return ActedMonoid::id();
        return prod_node(_root, 0, _n, l, r, ActedMonoid::op_id());
    }

    T all_prod() const { return _root ? (*_pool)[_root].val : ActedMonoid::id(); }

    std::vector<T> to_vector() const { return to_vector(0, _n); }

    std::vector<T> to_vector(int l, int r) const {
        assert(0 <= l && l <= r && r <= _n);
        std::vector<T> res;
        res.reserve(r - l);
        collect_node(_root, 0, _n, l, r, ActedMonoid::op_id(), res);
        return res;
    }

    PersistentLazySegtree apply(int p, const F& f) const {
        assert(0 <= p && p < _n);
        return apply(p, p + 1, f);
    }

    PersistentLazySegtree apply(int l, int r, const F& f) const {
        assert(0 <= l && l <= r && r <= _n);
        if (l == r) return *this;
        return PersistentLazySegtree(_n, apply_node(_root, 0, _n, l, r, f), _pool);
    }

    void apply_inplace(int p, const F& f) {
        assert(0 <= p && p < _n);
        apply_inplace(p, p + 1, f);
    }

    void apply_inplace(int l, int r, const F& f) {
        assert(0 <= l && l <= r && r <= _n);
        if (l == r) return;
        int root = apply_node(_root, 0, _n, l, r, f, true);
        _pool->replace(_root, root);
    }

    PersistentLazySegtree copy_range_from(const PersistentLazySegtree& source, int l, int r) const {
        assert(_n == source._n);
        assert(_pool == source._pool);
        assert(0 <= l && l <= r && r <= _n);
        if (l == r) return *this;
        int root = copy_range_node(_root, source._root, 0, _n, l, r);
        return PersistentLazySegtree(_n, root, _pool);
    }

    template <class G>
    int max_right(int l, G g) const {
        assert(0 <= l && l <= _n);
        assert(g(ActedMonoid::id()));
        if (l == _n) return _n;
        T sm = ActedMonoid::id();
        return max_right_node(_root, 0, _n, l, sm, ActedMonoid::op_id(), g);
    }

    template <class G>
    int min_left(int r, G g) const {
        assert(0 <= r && r <= _n);
        assert(g(ActedMonoid::id()));
        if (r == 0) return 0;
        T sm = ActedMonoid::id();
        return min_left_node(_root, 0, _n, r, sm, ActedMonoid::op_id(), g);
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "ds/segtree/persistent_segtree.hpp"



#line 9 "ds/segtree/persistent_segtree.hpp"

#line 12 "ds/segtree/persistent_segtree.hpp"

namespace m1une {
namespace ds {

template <m1une::monoid::IsMonoid Monoid>
struct PersistentSegtree {
    using T = typename Monoid::value_type;

   private:
    struct Node {
        T val;
        int left, right;
        int references;

        Node() : val(Monoid::id()), left(0), right(0), references(0) {}
        explicit Node(T value) : val(std::move(value)), left(0), right(0), references(0) {}
        Node(T value, int left_child, int right_child)
            : val(std::move(value)), left(left_child), right(right_child), references(0) {}
    };

    using Pool = detail::PersistentNodePool<Node>;

    int _n;
    int _root;
    std::shared_ptr<Pool> _pool;

    explicit PersistentSegtree(int n, int root, std::shared_ptr<Pool> pool)
        : _n(n), _root(root), _pool(std::move(pool)) {
        _pool->retain(_root);
    }

    int new_node(const Node& node) const { return _pool->emplace(node); }

    int new_node(Node&& node) const { return _pool->emplace(std::move(node)); }

    template <typename U>
    static T make_value(const U& value, int index) {
        if constexpr (requires(U x) { Monoid::make(x); }) {
            return Monoid::make(value);
        } else if constexpr (requires(U x, int i) { Monoid::make(x, i); }) {
            return Monoid::make(value, index);
        } else {
            return static_cast<T>(value);
        }
    }

    int build(int l, int r, const std::vector<T>& v) const {
        if (l == r) return 0;
        if (r - l == 1) return new_node(Node(v[l]));
        int m = (l + r) >> 1;
        int left = build(l, m, v);
        int right = build(m, r, v);
        return new_node(Node(Monoid::op(_pool->nodes[left].val, _pool->nodes[right].val), left, right));
    }

    int build(int l, int r, std::vector<T>& v) const {
        if (l == r) return 0;
        if (r - l == 1) return new_node(Node(std::move(v[l])));
        int m = (l + r) >> 1;
        int left = build(l, m, v);
        int right = build(m, r, v);
        return new_node(Node(Monoid::op(_pool->nodes[left].val, _pool->nodes[right].val), left, right));
    }

    template <typename U>
    int build_from_values(int l, int r, const std::vector<U>& v) const {
        if (l == r) return 0;
        if (r - l == 1) return new_node(Node(make_value(v[l], l)));
        int m = (l + r) >> 1;
        int left = build_from_values(l, m, v);
        int right = build_from_values(m, r, v);
        return new_node(Node(Monoid::op(_pool->nodes[left].val, _pool->nodes[right].val), left, right));
    }

    int set_node(int t, int l, int r, int p, T value, bool copy_on_write = false) const {
        if (copy_on_write) t = _pool->clone_if_shared(t);
        if (r - l == 1) {
            if (copy_on_write) {
                _pool->nodes[t].val = std::move(value);
                return t;
            }
            return new_node(Node(std::move(value)));
        }
        int m = (l + r) >> 1;
        int left = _pool->nodes[t].left;
        int right = _pool->nodes[t].right;
        if (p < m) {
            left = set_node(left, l, m, p, std::move(value), copy_on_write);
        } else {
            right = set_node(right, m, r, p, std::move(value), copy_on_write);
        }
        T product = Monoid::op(_pool->nodes[left].val, _pool->nodes[right].val);
        if (copy_on_write) {
            _pool->replace(_pool->nodes[t].left, left);
            _pool->replace(_pool->nodes[t].right, right);
            _pool->nodes[t].val = std::move(product);
            return t;
        }
        return new_node(Node(std::move(product), left, right));
    }

    T prod_node(int t, int l, int r, int ql, int qr) const {
        if (!t || qr <= l || r <= ql) return Monoid::id();
        if (ql <= l && r <= qr) return _pool->nodes[t].val;
        int m = (l + r) >> 1;
        return Monoid::op(prod_node(_pool->nodes[t].left, l, m, ql, qr),
                          prod_node(_pool->nodes[t].right, m, r, ql, qr));
    }

    void collect_node(int t, int l, int r, int ql, int qr, std::vector<T>& res) const {
        if (!t || qr <= l || r <= ql) return;
        if (r - l == 1) {
            res.push_back(_pool->nodes[t].val);
            return;
        }
        int m = (l + r) >> 1;
        collect_node(_pool->nodes[t].left, l, m, ql, qr, res);
        collect_node(_pool->nodes[t].right, m, r, ql, qr, res);
    }

    template <class F>
    int max_right_node(int t, int l, int r, int ql, T& sm, F& f) const {
        if (r <= ql) return r;
        if (ql <= l) {
            T nxt = Monoid::op(sm, _pool->nodes[t].val);
            if (f(nxt)) {
                sm = std::move(nxt);
                return r;
            }
            if (r - l == 1) return l;
        }
        int m = (l + r) >> 1;
        int res = max_right_node(_pool->nodes[t].left, l, m, ql, sm, f);
        if (res < m) return res;
        return max_right_node(_pool->nodes[t].right, m, r, ql, sm, f);
    }

    template <class F>
    int min_left_node(int t, int l, int r, int qr, T& sm, F& f) const {
        if (qr <= l) return l;
        if (r <= qr) {
            T nxt = Monoid::op(_pool->nodes[t].val, sm);
            if (f(nxt)) {
                sm = std::move(nxt);
                return l;
            }
            if (r - l == 1) return r;
        }
        int m = (l + r) >> 1;
        int res = min_left_node(_pool->nodes[t].right, m, r, qr, sm, f);
        if (m < res) return res;
        return min_left_node(_pool->nodes[t].left, l, m, qr, sm, f);
    }

   public:
    PersistentSegtree() : PersistentSegtree(0) {}

    explicit PersistentSegtree(int n) : _n(n), _root(0), _pool(std::make_shared<Pool>()) {
        assert(0 <= n);
        if (_n > 0) _root = build(0, _n, std::vector<T>(_n, Monoid::id()));
        _pool->retain(_root);
    }

    explicit PersistentSegtree(const std::vector<T>& v) : _n(int(v.size())), _root(0), _pool(std::make_shared<Pool>()) {
        _pool->reserve(v.size() * 2);
        if (_n > 0) _root = build(0, _n, v);
        _pool->retain(_root);
    }

    explicit PersistentSegtree(std::vector<T>&& v) : _n(int(v.size())), _root(0), _pool(std::make_shared<Pool>()) {
        _pool->reserve(v.size() * 2);
        if (_n > 0) _root = build(0, _n, v);
        _pool->retain(_root);
    }

    template <typename U>
        requires(!std::same_as<U, T>) &&
                (requires(U x) { Monoid::make(x); } || requires(U x, int i) { Monoid::make(x, i); } ||
                 std::convertible_to<U, T>)
    explicit PersistentSegtree(const std::vector<U>& v)
        : _n(int(v.size())), _root(0), _pool(std::make_shared<Pool>()) {
        _pool->reserve(v.size() * 2);
        if (_n > 0) _root = build_from_values(0, _n, v);
        _pool->retain(_root);
    }

    PersistentSegtree(const PersistentSegtree& other) : _n(other._n), _root(other._root), _pool(other._pool) {
        if (_pool) _pool->retain(_root);
    }

    PersistentSegtree(PersistentSegtree&& other) noexcept
        : _n(other._n), _root(other._root), _pool(std::move(other._pool)) {
        other._n = 0;
        other._root = 0;
    }

    PersistentSegtree& operator=(const PersistentSegtree& other) {
        if (this == &other) return *this;
        if (other._pool) other._pool->retain(other._root);
        if (_pool) _pool->release(_root);
        _n = other._n;
        _root = other._root;
        _pool = other._pool;
        return *this;
    }

    PersistentSegtree& operator=(PersistentSegtree&& other) noexcept {
        if (this == &other) return *this;
        if (_pool) _pool->release(_root);
        _n = other._n;
        _root = other._root;
        _pool = std::move(other._pool);
        other._n = 0;
        other._root = 0;
        return *this;
    }

    ~PersistentSegtree() {
        if (_pool) _pool->release(_root);
    }

    int size() const { return _n; }

    bool empty() const { return _n == 0; }

    // Drops this version immediately. Other versions and shared nodes stay valid.
    void release() {
        if (_pool) _pool->release(_root);
        _pool = std::make_shared<Pool>();
        _root = 0;
        _n = 0;
    }

    std::size_t node_count() const { return _pool ? _pool->size() : 0; }

    PersistentSegtree set(int p, T x) const {
        assert(0 <= p && p < _n);
        return PersistentSegtree(_n, set_node(_root, 0, _n, p, std::move(x)), _pool);
    }

    void set_inplace(int p, T x) {
        assert(0 <= p && p < _n);
        int root = set_node(_root, 0, _n, p, std::move(x), true);
        _pool->replace(_root, root);
    }

    T get(int p) const {
        assert(0 <= p && p < _n);
        int t = _root;
        int l = 0, r = _n;
        while (r - l > 1) {
            int m = (l + r) >> 1;
            if (p < m) {
                t = _pool->nodes[t].left;
                r = m;
            } else {
                t = _pool->nodes[t].right;
                l = m;
            }
        }
        return _pool->nodes[t].val;
    }

    T operator[](int p) const { return get(p); }

    T prod(int l, int r) const {
        assert(0 <= l && l <= r && r <= _n);
        if (l == r) return Monoid::id();
        return prod_node(_root, 0, _n, l, r);
    }

    T all_prod() const { return _root ? _pool->nodes[_root].val : Monoid::id(); }

    std::vector<T> to_vector() const { return to_vector(0, _n); }

    std::vector<T> to_vector(int l, int r) const {
        assert(0 <= l && l <= r && r <= _n);
        std::vector<T> res;
        res.reserve(r - l);
        collect_node(_root, 0, _n, l, r, res);
        return res;
    }

    template <class F>
    int max_right(int l, F f) const {
        assert(0 <= l && l <= _n);
        assert(f(Monoid::id()));
        if (l == _n) return _n;
        T sm = Monoid::id();
        return max_right_node(_root, 0, _n, l, sm, f);
    }

    template <class F>
    int min_left(int r, F f) const {
        assert(0 <= r && r <= _n);
        assert(f(Monoid::id()));
        if (r == 0) return 0;
        T sm = Monoid::id();
        return min_left_node(_root, 0, _n, r, sm, f);
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "ds/segtree/persistent_segtree_beats.hpp"



#line 10 "ds/segtree/persistent_segtree_beats.hpp"

#line 1 "beats_acted_monoid/concept.hpp"



#line 5 "beats_acted_monoid/concept.hpp"

#line 7 "beats_acted_monoid/concept.hpp"

namespace m1une {
namespace beats_acted_monoid {

// An acted monoid whose action may require descent before it can be applied.
template <typename AM>
concept IsBeatsActedMonoid = m1une::acted_monoid::IsActedMonoid<AM> &&
    requires(typename AM::value_type x, typename AM::operator_type f) {
        { AM::can_apply(f, x) } -> std::same_as<bool>;
    };

}  // namespace beats_acted_monoid
}  // namespace m1une


#line 13 "ds/segtree/persistent_segtree_beats.hpp"

namespace m1une {
namespace ds {

// A persistent Segment Tree Beats for fallible monoid actions.
template <m1une::beats_acted_monoid::IsBeatsActedMonoid ActedMonoid>
struct PersistentSegtreeBeats {
    using value_type = typename ActedMonoid::value_type;
    using operator_type = typename ActedMonoid::operator_type;
    using T = value_type;
    using F = operator_type;

   private:
    struct Node {
        T val;
        F lazy;
        int left;
        int right;
        int references;
        bool has_lazy;

        Node()
            : val(ActedMonoid::id()),
              lazy(ActedMonoid::op_id()),
              left(0),
              right(0),
              references(0),
              has_lazy(false) {}

        explicit Node(T value)
            : val(std::move(value)),
              lazy(ActedMonoid::op_id()),
              left(0),
              right(0),
              references(0),
              has_lazy(false) {}

        Node(T value, int left_child, int right_child)
            : val(std::move(value)),
              lazy(ActedMonoid::op_id()),
              left(left_child),
              right(right_child),
              references(0),
              has_lazy(false) {}
    };

    using Pool = detail::PersistentNodePool<Node>;

    int _n;
    int _root;
    std::shared_ptr<Pool> _pool;

    explicit PersistentSegtreeBeats(
        int n,
        int root,
        std::shared_ptr<Pool> pool
    ) : _n(n), _root(root), _pool(std::move(pool)) {
        _pool->retain(_root);
    }

    int new_node(const Node& node) const {
        return _pool->emplace(node);
    }

    int new_node(Node&& node) const {
        return _pool->emplace(std::move(node));
    }

    int clone_node(int node) const {
        return _pool->clone(node);
    }

    template <typename U>
    static T make_value(const U& value, int index) {
        if constexpr (requires(U x) { ActedMonoid::make(x); }) {
            return ActedMonoid::make(value);
        } else if constexpr (requires(U x, int i) {
            ActedMonoid::make(x, i);
        }) {
            return ActedMonoid::make(value, index);
        } else {
            return static_cast<T>(value);
        }
    }

    static T mapping_at(const F& f, const T& value, long long ordinal) {
        if constexpr (requires(F g, T x, long long i) {
            ActedMonoid::mapping(g, x, i);
        }) {
            return ActedMonoid::mapping(f, value, ordinal);
        } else {
            return ActedMonoid::mapping(f, value);
        }
    }

    static bool can_apply_at(
        const F& f,
        const T& value,
        long long ordinal
    ) {
        if constexpr (requires(F g, T x, long long i) {
            ActedMonoid::can_apply(g, x, i);
        }) {
            return ActedMonoid::can_apply(f, value, ordinal);
        } else {
            return ActedMonoid::can_apply(f, value);
        }
    }

    static F shift_operator(const F& f, long long ordinal) {
        if constexpr (requires(F g, long long i) {
            ActedMonoid::op_shift(g, i);
        }) {
            return ActedMonoid::op_shift(f, ordinal);
        } else {
            return f;
        }
    }

    int build(int left, int right, const std::vector<T>& values) const {
        if (left == right) return 0;
        if (right - left == 1) return new_node(Node(values[left]));
        int middle = left + (right - left) / 2;
        int left_child = build(left, middle, values);
        int right_child = build(middle, right, values);
        return new_node(Node(
            ActedMonoid::op(
                (*_pool)[left_child].val,
                (*_pool)[right_child].val
            ),
            left_child,
            right_child
        ));
    }

    int build(int left, int right, std::vector<T>& values) const {
        if (left == right) return 0;
        if (right - left == 1) {
            return new_node(Node(std::move(values[left])));
        }
        int middle = left + (right - left) / 2;
        int left_child = build(left, middle, values);
        int right_child = build(middle, right, values);
        return new_node(Node(
            ActedMonoid::op(
                (*_pool)[left_child].val,
                (*_pool)[right_child].val
            ),
            left_child,
            right_child
        ));
    }

    template <typename U>
    int build_from_values(
        int left,
        int right,
        const std::vector<U>& values
    ) const {
        if (left == right) return 0;
        if (right - left == 1) {
            return new_node(Node(make_value(values[left], left)));
        }
        int middle = left + (right - left) / 2;
        int left_child = build_from_values(left, middle, values);
        int right_child = build_from_values(middle, right, values);
        return new_node(Node(
            ActedMonoid::op(
                (*_pool)[left_child].val,
                (*_pool)[right_child].val
            ),
            left_child,
            right_child
        ));
    }

    void update(int node) const {
        Node& current = (*_pool)[node];
        current.val = ActedMonoid::op(
            (*_pool)[current.left].val,
            (*_pool)[current.right].val
        );
    }

    int all_apply_clone(
        int node,
        int left,
        int right,
        const F& f,
        bool copy_on_write = false
    ) const {
        int result = copy_on_write ? _pool->clone_if_shared(node) : clone_node(node);
        Node& current = (*_pool)[result];
        if (can_apply_at(f, current.val, 0)) {
            current.val = mapping_at(f, current.val, 0);
            if (right - left > 1) {
                current.lazy = ActedMonoid::op_comp(f, current.lazy);
                current.has_lazy = true;
            }
            return result;
        }

        assert(right - left > 1);
        push(result, left, right, copy_on_write);
        int middle = left + (right - left) / 2;
        int left_child = all_apply_clone(
            (*_pool)[result].left,
            left,
            middle,
            f,
            copy_on_write
        );
        int right_child = all_apply_clone(
            (*_pool)[result].right,
            middle,
            right,
            shift_operator(f, middle - left),
            copy_on_write
        );
        _pool->replace((*_pool)[result].left, left_child);
        _pool->replace((*_pool)[result].right, right_child);
        update(result);
        return result;
    }

    void push(
        int node,
        int left,
        int right,
        bool copy_on_write = false
    ) const {
        if (!(*_pool)[node].has_lazy) return;
        assert(right - left > 1);

        F lazy = (*_pool)[node].lazy;
        int middle = left + (right - left) / 2;
        int left_child = all_apply_clone(
            (*_pool)[node].left,
            left,
            middle,
            lazy,
            copy_on_write
        );
        int right_child = all_apply_clone(
            (*_pool)[node].right,
            middle,
            right,
            shift_operator(lazy, middle - left),
            copy_on_write
        );
        _pool->replace((*_pool)[node].left, left_child);
        _pool->replace((*_pool)[node].right, right_child);
        Node& current = (*_pool)[node];
        current.lazy = ActedMonoid::op_id();
        current.has_lazy = false;
    }

    int set_node(
        int node,
        int left,
        int right,
        int index,
        T value,
        bool copy_on_write = false
    ) const {
        int result = copy_on_write ? _pool->clone_if_shared(node) : clone_node(node);
        if (right - left == 1) {
            Node& current = (*_pool)[result];
            current.val = std::move(value);
            current.lazy = ActedMonoid::op_id();
            current.has_lazy = false;
            return result;
        }

        push(result, left, right, copy_on_write);
        int middle = left + (right - left) / 2;
        if (index < middle) {
            int child = set_node(
                (*_pool)[result].left,
                left,
                middle,
                index,
                std::move(value),
                copy_on_write
            );
            _pool->replace((*_pool)[result].left, child);
        } else {
            int child = set_node(
                (*_pool)[result].right,
                middle,
                right,
                index,
                std::move(value),
                copy_on_write
            );
            _pool->replace((*_pool)[result].right, child);
        }
        update(result);
        return result;
    }

    int apply_node(
        int node,
        int left,
        int right,
        int query_left,
        int query_right,
        const F& f,
        bool copy_on_write = false
    ) const {
        if (query_right <= left || right <= query_left) return node;
        if (query_left <= left && right <= query_right) {
            return all_apply_clone(
                node,
                left,
                right,
                shift_operator(f, left - query_left),
                copy_on_write
            );
        }

        int result = copy_on_write ? _pool->clone_if_shared(node) : clone_node(node);
        push(result, left, right, copy_on_write);
        int middle = left + (right - left) / 2;
        int left_child = apply_node(
            (*_pool)[result].left,
            left,
            middle,
            query_left,
            query_right,
            f,
            copy_on_write
        );
        int right_child = apply_node(
            (*_pool)[result].right,
            middle,
            right,
            query_left,
            query_right,
            f,
            copy_on_write
        );
        _pool->replace((*_pool)[result].left, left_child);
        _pool->replace((*_pool)[result].right, right_child);
        update(result);
        return result;
    }

    int copy_range_node(
        int target,
        int source,
        int left,
        int right,
        int query_left,
        int query_right
    ) const {
        if (query_right <= left || right <= query_left) return target;
        if (query_left <= left && right <= query_right) return source;

        int result = clone_node(target);
        int materialized_source = clone_node(source);
        _pool->retain(materialized_source);
        push(result, left, right);
        push(materialized_source, left, right);

        int middle = left + (right - left) / 2;
        int left_child = copy_range_node(
            (*_pool)[result].left,
            (*_pool)[materialized_source].left,
            left,
            middle,
            query_left,
            query_right
        );
        int right_child = copy_range_node(
            (*_pool)[result].right,
            (*_pool)[materialized_source].right,
            middle,
            right,
            query_left,
            query_right
        );
        _pool->replace((*_pool)[result].left, left_child);
        _pool->replace((*_pool)[result].right, right_child);
        update(result);
        _pool->release(materialized_source);
        return result;
    }

    F compose_for_child(
        const F& inherited,
        const Node& node,
        long long ordinal
    ) const {
        F shifted = shift_operator(inherited, ordinal);
        if (!node.has_lazy) return shifted;
        return ActedMonoid::op_comp(
            shifted,
            shift_operator(node.lazy, ordinal)
        );
    }

    T evaluate_node(
        int node,
        int left,
        int right,
        const F& inherited
    ) const {
        const Node& current = (*_pool)[node];
        if (can_apply_at(inherited, current.val, 0)) {
            return mapping_at(inherited, current.val, 0);
        }

        assert(right - left > 1);
        int middle = left + (right - left) / 2;
        return ActedMonoid::op(
            evaluate_node(
                current.left,
                left,
                middle,
                compose_for_child(inherited, current, 0)
            ),
            evaluate_node(
                current.right,
                middle,
                right,
                compose_for_child(inherited, current, middle - left)
            )
        );
    }

    T prod_node(
        int node,
        int left,
        int right,
        int query_left,
        int query_right,
        const F& inherited
    ) const {
        if (query_right <= left || right <= query_left) {
            return ActedMonoid::id();
        }
        if (query_left <= left && right <= query_right) {
            return evaluate_node(node, left, right, inherited);
        }

        const Node& current = (*_pool)[node];
        int middle = left + (right - left) / 2;
        return ActedMonoid::op(
            prod_node(
                current.left,
                left,
                middle,
                query_left,
                query_right,
                compose_for_child(inherited, current, 0)
            ),
            prod_node(
                current.right,
                middle,
                right,
                query_left,
                query_right,
                compose_for_child(inherited, current, middle - left)
            )
        );
    }

    void collect_node(
        int node,
        int left,
        int right,
        int query_left,
        int query_right,
        const F& inherited,
        std::vector<T>& result
    ) const {
        if (query_right <= left || right <= query_left) return;
        const Node& current = (*_pool)[node];
        if (right - left == 1) {
            result.push_back(mapping_at(inherited, current.val, 0));
            return;
        }

        int middle = left + (right - left) / 2;
        collect_node(
            current.left,
            left,
            middle,
            query_left,
            query_right,
            compose_for_child(inherited, current, 0),
            result
        );
        collect_node(
            current.right,
            middle,
            right,
            query_left,
            query_right,
            compose_for_child(inherited, current, middle - left),
            result
        );
    }

    template <class Predicate>
    int max_right_node(
        int node,
        int left,
        int right,
        int query_left,
        T& product,
        const F& inherited,
        Predicate& predicate
    ) const {
        if (right <= query_left) return right;
        if (query_left <= left) {
            T next = ActedMonoid::op(
                product,
                evaluate_node(node, left, right, inherited)
            );
            if (predicate(next)) {
                product = std::move(next);
                return right;
            }
            if (right - left == 1) return left;
        }

        const Node& current = (*_pool)[node];
        int middle = left + (right - left) / 2;
        int result = max_right_node(
            current.left,
            left,
            middle,
            query_left,
            product,
            compose_for_child(inherited, current, 0),
            predicate
        );
        if (result < middle) return result;
        return max_right_node(
            current.right,
            middle,
            right,
            query_left,
            product,
            compose_for_child(inherited, current, middle - left),
            predicate
        );
    }

    template <class Predicate>
    int min_left_node(
        int node,
        int left,
        int right,
        int query_right,
        T& product,
        const F& inherited,
        Predicate& predicate
    ) const {
        if (query_right <= left) return left;
        if (right <= query_right) {
            T next = ActedMonoid::op(
                evaluate_node(node, left, right, inherited),
                product
            );
            if (predicate(next)) {
                product = std::move(next);
                return left;
            }
            if (right - left == 1) return right;
        }

        const Node& current = (*_pool)[node];
        int middle = left + (right - left) / 2;
        int result = min_left_node(
            current.right,
            middle,
            right,
            query_right,
            product,
            compose_for_child(inherited, current, middle - left),
            predicate
        );
        if (middle < result) return result;
        return min_left_node(
            current.left,
            left,
            middle,
            query_right,
            product,
            compose_for_child(inherited, current, 0),
            predicate
        );
    }

   public:
    PersistentSegtreeBeats() : PersistentSegtreeBeats(0) {}

    explicit PersistentSegtreeBeats(int n)
        : _n(n), _root(0), _pool(std::make_shared<Pool>()) {
        assert(0 <= n);
        if (_n > 0) {
            std::vector<T> values(_n, ActedMonoid::id());
            _root = build(0, _n, values);
        }
        _pool->retain(_root);
    }

    explicit PersistentSegtreeBeats(const std::vector<T>& values)
        : _n(int(values.size())),
          _root(0),
          _pool(std::make_shared<Pool>()) {
        _pool->reserve(values.size() * 2);
        if (_n > 0) _root = build(0, _n, values);
        _pool->retain(_root);
    }

    explicit PersistentSegtreeBeats(std::vector<T>&& values)
        : _n(int(values.size())),
          _root(0),
          _pool(std::make_shared<Pool>()) {
        _pool->reserve(values.size() * 2);
        if (_n > 0) _root = build(0, _n, values);
        _pool->retain(_root);
    }

    template <typename U>
    requires (!std::same_as<U, T>) && (
        requires(U x) { ActedMonoid::make(x); } ||
        requires(U x, int i) { ActedMonoid::make(x, i); } ||
        std::convertible_to<U, T>
    )
    explicit PersistentSegtreeBeats(const std::vector<U>& values)
        : _n(int(values.size())),
          _root(0),
          _pool(std::make_shared<Pool>()) {
        _pool->reserve(values.size() * 2);
        if (_n > 0) _root = build_from_values(0, _n, values);
        _pool->retain(_root);
    }

    PersistentSegtreeBeats(const PersistentSegtreeBeats& other)
        : _n(other._n), _root(other._root), _pool(other._pool) {
        if (_pool) _pool->retain(_root);
    }

    PersistentSegtreeBeats(PersistentSegtreeBeats&& other) noexcept
        : _n(other._n),
          _root(other._root),
          _pool(std::move(other._pool)) {
        other._n = 0;
        other._root = 0;
    }

    PersistentSegtreeBeats& operator=(
        const PersistentSegtreeBeats& other
    ) {
        if (this == &other) return *this;
        if (other._pool) other._pool->retain(other._root);
        if (_pool) _pool->release(_root);
        _n = other._n;
        _root = other._root;
        _pool = other._pool;
        return *this;
    }

    PersistentSegtreeBeats& operator=(
        PersistentSegtreeBeats&& other
    ) noexcept {
        if (this == &other) return *this;
        if (_pool) _pool->release(_root);
        _n = other._n;
        _root = other._root;
        _pool = std::move(other._pool);
        other._n = 0;
        other._root = 0;
        return *this;
    }

    ~PersistentSegtreeBeats() {
        if (_pool) _pool->release(_root);
    }

    int size() const {
        return _n;
    }

    bool empty() const {
        return _n == 0;
    }

    void release() {
        if (_pool) _pool->release(_root);
        _pool = std::make_shared<Pool>();
        _root = 0;
        _n = 0;
    }

    std::size_t node_count() const {
        return _pool ? _pool->size() : 0;
    }

    PersistentSegtreeBeats set(int index, T value) const {
        assert(0 <= index && index < _n);
        return PersistentSegtreeBeats(
            _n,
            set_node(_root, 0, _n, index, std::move(value)),
            _pool
        );
    }

    void set_inplace(int index, T value) {
        assert(0 <= index && index < _n);
        int root = set_node(
            _root,
            0,
            _n,
            index,
            std::move(value),
            true
        );
        _pool->replace(_root, root);
    }

    T get(int index) const {
        assert(0 <= index && index < _n);
        return prod(index, index + 1);
    }

    T operator[](int index) const {
        return get(index);
    }

    T prod(int left, int right) const {
        assert(0 <= left && left <= right && right <= _n);
        if (left == right) return ActedMonoid::id();
        return prod_node(
            _root,
            0,
            _n,
            left,
            right,
            ActedMonoid::op_id()
        );
    }

    T all_prod() const {
        return _root ? (*_pool)[_root].val : ActedMonoid::id();
    }

    PersistentSegtreeBeats apply(int index, const F& f) const {
        assert(0 <= index && index < _n);
        return apply(index, index + 1, f);
    }

    PersistentSegtreeBeats apply(
        int left,
        int right,
        const F& f
    ) const {
        assert(0 <= left && left <= right && right <= _n);
        if (left == right) return *this;
        return PersistentSegtreeBeats(
            _n,
            apply_node(_root, 0, _n, left, right, f),
            _pool
        );
    }

    void apply_inplace(int index, const F& f) {
        assert(0 <= index && index < _n);
        apply_inplace(index, index + 1, f);
    }

    void apply_inplace(int left, int right, const F& f) {
        assert(0 <= left && left <= right && right <= _n);
        if (left == right) return;
        int root = apply_node(
            _root,
            0,
            _n,
            left,
            right,
            f,
            true
        );
        _pool->replace(_root, root);
    }

    PersistentSegtreeBeats copy_range_from(
        const PersistentSegtreeBeats& source,
        int left,
        int right
    ) const {
        assert(_n == source._n);
        assert(_pool == source._pool);
        assert(0 <= left && left <= right && right <= _n);
        if (left == right) return *this;
        return PersistentSegtreeBeats(
            _n,
            copy_range_node(
                _root,
                source._root,
                0,
                _n,
                left,
                right
            ),
            _pool
        );
    }

    std::vector<T> to_vector() const {
        return to_vector(0, _n);
    }

    std::vector<T> to_vector(int left, int right) const {
        assert(0 <= left && left <= right && right <= _n);
        std::vector<T> result;
        result.reserve(right - left);
        if (left != right) {
            collect_node(
                _root,
                0,
                _n,
                left,
                right,
                ActedMonoid::op_id(),
                result
            );
        }
        return result;
    }

    template <class Predicate>
    int max_right(int left, Predicate predicate) const {
        assert(0 <= left && left <= _n);
        assert(predicate(ActedMonoid::id()));
        if (left == _n) return _n;
        T product = ActedMonoid::id();
        return max_right_node(
            _root,
            0,
            _n,
            left,
            product,
            ActedMonoid::op_id(),
            predicate
        );
    }

    template <class Predicate>
    int min_left(int right, Predicate predicate) const {
        assert(0 <= right && right <= _n);
        assert(predicate(ActedMonoid::id()));
        if (right == 0) return 0;
        T product = ActedMonoid::id();
        return min_left_node(
            _root,
            0,
            _n,
            right,
            product,
            ActedMonoid::op_id(),
            predicate
        );
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "monoid/add.hpp"



namespace m1une {
namespace monoid {

// Monoid for addition (Range Sum).
template <typename T>
struct Add {
    using value_type = T;
    static constexpr bool commutative = true;

    // Returns the identity element for addition, which is 0.
    static constexpr T id() {
        return T(0);
    }

    // Returns the sum of a and b.
    static constexpr T op(const T& a, const T& b) {
        return a + b;
    }

    static constexpr T inv(const T& x) {
        return -x;
    }
};

}  // namespace monoid
}  // namespace m1une


#line 1 "utilities/fast_io.hpp"



#line 5 "utilities/fast_io.hpp"
#include <array>
#include <cerrno>
#include <charconv>
#line 9 "utilities/fast_io.hpp"
#include <cstdio>
#include <cstdlib>
#line 12 "utilities/fast_io.hpp"
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#line 18 "utilities/fast_io.hpp"
#include <unistd.h>
#line 20 "utilities/fast_io.hpp"

namespace m1une {
namespace utilities {

struct FastOutput;

namespace internal {

// Shared with the convenience helpers in template.hpp.
inline FastOutput* standard_output_instance = nullptr;

// Detect std::begin(x), std::end(x).
template <class T, class = void>
struct is_range : std::false_type {};

template <class T>
struct is_range<T, std::void_t<
    decltype(std::begin(std::declval<T&>())),
    decltype(std::end(std::declval<T&>()))
>> : std::true_type {};

template <class T>
inline constexpr bool is_range_v = is_range<T>::value;

template <class T>
using range_reference_t = decltype(*std::begin(std::declval<T&>()));

template <class T>
using range_value_t = std::remove_cv_t<std::remove_reference_t<range_reference_t<T>>>;

template <class T, class = void>
struct range_stored_value {
    using type = range_value_t<T>;
};

template <class T>
struct range_stored_value<T, std::void_t<typename std::remove_cv_t<std::remove_reference_t<T>>::value_type>> {
    using type = typename std::remove_cv_t<std::remove_reference_t<T>>::value_type;
};

template <class T>
using range_stored_value_t = typename range_stored_value<T>::type;

// Treat strings and C strings as scalar output objects, not as ranges.
template <class T>
struct is_char_array : std::false_type {};

template <class T, std::size_t N>
struct is_char_array<T[N]>
    : std::bool_constant<std::is_same_v<std::remove_cv_t<T>, char>> {};

template <class T>
struct is_string_like
    : std::bool_constant<
          std::is_same_v<std::decay_t<T>, std::string>
          || std::is_same_v<std::decay_t<T>, const char*>
          || std::is_same_v<std::decay_t<T>, char*>
          || is_char_array<std::remove_reference_t<T>>::value
      > {};

template <class T>
inline constexpr bool is_string_like_v = is_string_like<T>::value;

// ModInt-like type: x.val() is printable, and x can be assigned from long long.
template <class T, class = void>
struct has_val_method : std::false_type {};

template <class T>
struct has_val_method<T, std::void_t<decltype(std::declval<const T&>().val())>>
    : std::true_type {};

template <class T>
inline constexpr bool has_val_method_v = has_val_method<T>::value;

template <class T, class = void>
struct has_static_mod_raw : std::false_type {};

template <class T>
struct has_static_mod_raw<
    T, std::void_t<decltype(T::mod()), decltype(T::raw(std::declval<uint32_t>()))>>
    : std::true_type {};

template <class T>
inline constexpr bool has_static_mod_raw_v = has_static_mod_raw<T>::value;

// libstdc++ before GCC 16 does not classify __int128 as an integral type in
// strict ISO modes such as -std=c++23. Keep the fast-I/O interface independent
// of that implementation detail.
template <class T>
inline constexpr bool is_integral_v =
    std::is_integral_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>
    || std::is_same_v<std::remove_cv_t<T>, __uint128_t>;

template <class T>
inline constexpr bool is_signed_v =
    std::is_signed_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>;

template <class T>
struct make_unsigned {
    using type = std::make_unsigned_t<T>;
};

template <>
struct make_unsigned<__int128_t> {
    using type = __uint128_t;
};

template <>
struct make_unsigned<__uint128_t> {
    using type = __uint128_t;
};

template <class T>
using make_unsigned_t = typename make_unsigned<std::remove_cv_t<T>>::type;

}  // namespace internal

struct FastInput {
    static constexpr int buffer_size = 1 << 20;

   private:
    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _length;
    int _file_descriptor;
    bool _streaming;

    bool refill() {
        _position = 0;
        if (_streaming) {
            ssize_t length;
            do {
                length = ::read(_file_descriptor, _buffer, buffer_size);
            } while (length < 0 && errno == EINTR);
            if (length <= 0) {
                _length = 0;
                return false;
            }
            _length = int(length);
        } else {
            _length = int(std::fread(_buffer, 1, buffer_size, _stream));
        }
        return _length != 0;
    }

    template <class T>
    bool read_integer_from_stream(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = read_char_raw();
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = negative ? result * 10 - (c - '0')
                                  : result * 10 + (c - '0');
                c = read_char_raw();
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = result * 10 + T(c - '0');
                c = read_char_raw();
            }
            value = negative ? T(0) - result : result;
        }
        return true;
    }

    bool prepare_number() {
        if (_length - _position >= 64) return true;
        const int remaining = _length - _position;
        if (remaining > 0) std::memmove(_buffer, _buffer + _position, remaining);
        const int added = int(std::fread(_buffer + remaining, 1, buffer_size - remaining, _stream));
        _position = 0;
        _length = remaining + added;
        if (_length < buffer_size) _buffer[_length] = '\0';
        return _length != 0;
    }

   public:
    explicit FastInput(std::FILE* stream = stdin)
        : _stream(stream),
          _position(0),
          _length(0),
          _file_descriptor(::fileno(stream)),
          _streaming([&] {
              struct stat status;
              return _file_descriptor >= 0
                     && ::fstat(_file_descriptor, &status) == 0
                     && !S_ISREG(status.st_mode);
          }()) {}

    FastInput(const FastInput&) = delete;
    FastInput& operator=(const FastInput&) = delete;

    int read_char_raw() {
        if (_position == _length && !refill()) return EOF;
        return _buffer[_position++];
    }

    bool skip_spaces() {
        int c = read_char_raw();
        while (c != EOF && c <= ' ') c = read_char_raw();
        if (c == EOF) return false;
        --_position;
        return true;
    }

    bool read(char& value) {
        if (!skip_spaces()) return false;
        value = char(read_char_raw());
        return true;
    }

    bool read(std::string& value) {
        if (!skip_spaces()) return false;
        value.clear();
        while (true) {
            const int begin = _position;
            while (_position < _length &&
                   static_cast<unsigned char>(_buffer[_position]) > ' ') {
                ++_position;
            }
            value.append(_buffer + begin, _position - begin);
            if (_position < _length) {
                ++_position;
                return true;
            }
            if (!refill()) return true;
        }
    }

    bool read(bool& value) {
        int x;
        if (!read(x)) return false;
        value = x != 0;
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>,
        bool
    >
    read(T& value) {
        if (_streaming) return read_integer_from_stream(value);
        if (!prepare_number()) return false;
        int c = static_cast<unsigned char>(_buffer[_position++]);
        while (c <= ' ') c = static_cast<unsigned char>(_buffer[_position++]);

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = static_cast<unsigned char>(_buffer[_position++]);
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const int first = c - '0';
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = negative ? result * 100 - (first * 10 + second)
                                      : result * 100 + (first * 10 + second);
                    ++_position;
                } else {
                    result = negative ? result * 10 - first : result * 10 + first;
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const unsigned first = unsigned(c - '0');
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = result * 100 + T(first * 10 + unsigned(second));
                    ++_position;
                } else {
                    result = result * 10 + T(first);
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = negative ? T(0) - result : result;
        }
        if (_position > _length) _position = _length;
        return true;
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>, bool>
    read(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();
        bool negative = false;
        if (c == '-' || c == '+') {
            negative = c == '-';
            c = read_char_raw();
        }

        long double result = 0;
        while ('0' <= c && c <= '9') {
            result = result * 10 + (c - '0');
            c = read_char_raw();
        }
        if (c == '.') {
            long double place = 0.1L;
            c = read_char_raw();
            while ('0' <= c && c <= '9') {
                result += (c - '0') * place;
                place *= 0.1L;
                c = read_char_raw();
            }
        }
        if (c == 'e' || c == 'E') {
            c = read_char_raw();
            bool exponent_negative = false;
            if (c == '-' || c == '+') {
                exponent_negative = c == '-';
                c = read_char_raw();
            }
            int exponent = 0;
            while ('0' <= c && c <= '9') {
                exponent = exponent * 10 + (c - '0');
                c = read_char_raw();
            }
            long double scale = 1;
            long double power = 10;
            while (exponent > 0) {
                if (exponent & 1) scale *= power;
                power *= power;
                exponent >>= 1;
            }
            result = exponent_negative ? result / scale : result * scale;
        }
        value = static_cast<T>(negative ? -result : result);
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>,
        bool
    >
    read(T& value) {
        long long x;
        if (!read(x)) return false;
        if constexpr (internal::has_static_mod_raw_v<T>) {
            if (x >= 0 && uint64_t(x) < uint64_t(T::mod())) {
                value = T::raw(uint32_t(x));
            } else {
                value = T(x);
            }
        } else {
            value = T(x);
        }
        return true;
    }

    template <class First, class Second>
    bool read(std::pair<First, Second>& value) {
        if (!read(value.first)) return false;
        return read(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>,
        bool
    >
    read(Range& range) {
        using StoredValue = internal::range_stored_value_t<Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        for (auto&& value : range) {
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                bool x;
                if (!read(x)) return false;
                value = x;
            } else {
                if (!read(value)) return false;
            }
        }
        return true;
    }

    template <class First, class Second, class... Rest>
    bool read(First& first, Second& second, Rest&... rest) {
        if (!read(first)) return false;
        return read(second, rest...);
    }

    template <class T>
    FastInput& operator>>(T& value) {
        if (!read(value)) std::abort();
        return *this;
    }
};

struct FastOutput {
    static constexpr int buffer_size = 1 << 20;

   private:
    inline static const auto digit_quads = [] {
        std::array<char, 40000> result{};
        for (int i = 0; i < 10000; i++) {
            int value = i;
            for (int j = 3; j >= 0; j--) {
                result[4 * i + j] = char('0' + value % 10);
                value /= 10;
            }
        }
        return result;
    }();

    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _precision;
    std::chars_format _float_format;
    char _range_separator;
    std::string* _capture = nullptr;

    template <class T>
    std::string format_cell(const T& value) {
        std::string result;
        struct CaptureGuard {
            std::string*& target;
            std::string* previous;
            ~CaptureGuard() { target = previous; }
        } guard{_capture, _capture};
        _capture = &result;
        write(value);
        return result;
    }

    template <class Matrix>
    void write_aligned_matrix(const Matrix& matrix) {
        std::vector<std::vector<std::string>> rows;
        std::vector<std::size_t> widths;
        for (const auto& row : matrix) {
            auto& cells = rows.emplace_back();
            std::size_t column = 0;
            for (const auto& value : row) {
                cells.push_back(format_cell(value));
                if (column == widths.size()) widths.push_back(0);
                widths[column] = std::max(widths[column], cells.back().size());
                ++column;
            }
        }
        bool first = true;
        for (const auto& row : rows) {
            if (!first) write_char('\n');
            first = false;
            for (std::size_t column = 0; column < row.size(); ++column) {
                if (column != 0) write_char(_range_separator);
                for (std::size_t padding = row[column].size();
                     padding < widths[column]; ++padding) {
                    write_char(' ');
                }
                write(row[column]);
            }
        }
    }

   public:
    explicit FastOutput(std::FILE* stream = stdout)
        : _stream(stream),
          _position(0),
          _precision(6),
          _float_format(std::chars_format::general),
          _range_separator(' ') {
        if (_stream == stdout
            && internal::standard_output_instance == nullptr) {
            internal::standard_output_instance = this;
        }
    }

    FastOutput(const FastOutput&) = delete;
    FastOutput& operator=(const FastOutput&) = delete;

    ~FastOutput() {
        flush();
        if (internal::standard_output_instance == this) {
            internal::standard_output_instance = nullptr;
        }
    }

    void flush() {
        if (_position != 0) {
            std::fwrite(_buffer, 1, _position, _stream);
            _position = 0;
        }
        std::fflush(_stream);
    }

    void write_char(char c) {
        if (_capture != nullptr) {
            _capture->push_back(c);
            return;
        }
        if (_position == buffer_size) flush();
        _buffer[_position++] = c;
    }

    void write(const char* s) {
        while (*s != '\0') write_char(*s++);
    }

    void write(const std::string& s) {
        if (_capture != nullptr) {
            _capture->append(s);
            return;
        }
        std::size_t position = 0;
        while (position < s.size()) {
            if (_position == buffer_size) flush();
            const std::size_t copied =
                std::min<std::size_t>(buffer_size - _position, s.size() - position);
            std::memcpy(_buffer + _position, s.data() + position, copied);
            _position += int(copied);
            position += copied;
        }
    }

    void write(char c) {
        write_char(c);
    }

    void write(bool value) {
        write_char(value ? '1' : '0');
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>>
    write(T value) {
        char digits[128];
        auto [end, error] = std::to_chars(
            digits,
            digits + sizeof(digits),
            value,
            _float_format,
            _precision
        );
        if (error != std::errc()) std::abort();
        for (const char* pointer = digits; pointer != end; pointer++) {
            write_char(*pointer);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>
    >
    write(T value) {
        using Raw = std::remove_cv_t<T>;
        using Unsigned = internal::make_unsigned_t<Raw>;

        Unsigned magnitude;
        if constexpr (internal::is_signed_v<Raw>) {
            if (value < 0) {
                write_char('-');
                magnitude = Unsigned(0) - Unsigned(value);
            } else {
                magnitude = Unsigned(value);
            }
        } else {
            magnitude = value;
        }

        if (magnitude == 0) {
            write_char('0');
            return;
        }

        unsigned chunks[16];
        int count = 0;
        while (magnitude >= 10000) {
            const Unsigned quotient = magnitude / 10000;
            chunks[count++] = unsigned(magnitude - quotient * 10000);
            magnitude = quotient;
        }
        if (_capture == nullptr && _position > buffer_size - 64) flush();
        char captured[64];
        char* const begin = _capture != nullptr ? captured : _buffer + _position;
        char* destination = begin;
        const unsigned leading = unsigned(magnitude);
        const char* first = digit_quads.data() + 4 * leading;
        int skip = leading < 10 ? 3 : leading < 100 ? 2 : leading < 1000 ? 1 : 0;
        for (; skip < 4; skip++) *destination++ = first[skip];
        while (count--) {
            const char* digits = digit_quads.data() + 4 * chunks[count];
            std::memcpy(destination, digits, 4);
            destination += 4;
        }
        if (_capture != nullptr) {
            _capture->append(begin, destination - begin);
        } else {
            _position += int(destination - begin);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>
    >
    write(const T& value) {
        write(value.val());
    }

    template <class First, class Second>
    void write(const std::pair<First, Second>& value) {
        write(value.first);
        write_char(' ');
        write(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>
    >
    write(const Range& range) {
        using StoredValue = internal::range_stored_value_t<const Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        bool first = true;
        for (const auto& value : range) {
            if (!first) write_char(nested ? '\n' : _range_separator);
            first = false;
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                write(static_cast<bool>(value));
            } else {
                write(value);
            }
        }
    }

    template <class First, class... Rest>
    void print(const First& first, const Rest&... rest) {
        write(first);
        ((write_char(' '), write(rest)), ...);
    }

    void println() {
        write_char('\n');
    }

    void set_precision(int precision) {
        _precision = precision;
    }

    void set_fixed(int precision = 6) {
        _float_format = std::chars_format::fixed;
        _precision = precision;
    }

    void set_general(int precision = 6) {
        _float_format = std::chars_format::general;
        _precision = precision;
    }

    void set_range_separator(char separator) {
        _range_separator = separator;
    }

    template <class Matrix>
    void write_aligned(const Matrix& matrix) {
        using Row = internal::range_stored_value_t<const Matrix>;
        using Cell = internal::range_stored_value_t<const Row>;
        static_assert(internal::is_range_v<Row> && !internal::is_string_like_v<Row>,
                      "write_aligned requires a two-dimensional range");
        static_assert(!internal::is_range_v<Cell> || internal::is_string_like_v<Cell>,
                      "write_aligned requires scalar cells");
        write_aligned_matrix(matrix);
    }

    template <class Matrix>
    void println_aligned(const Matrix& matrix) {
        write_aligned(matrix);
        write_char('\n');
    }

    template <class... Args>
    void println(const Args&... args) {
        print(args...);
        write_char('\n');
    }

    template <class T>
    FastOutput& operator<<(const T& value) {
        write(value);
        return *this;
    }
};

}  // namespace utilities
}  // namespace m1une


#line 19 "verify/ds/persistent_cow.test.cpp"

#line 24 "verify/ds/persistent_cow.test.cpp"

namespace {

using Add = m1une::monoid::Add<long long>;
using RangeAdd = m1une::acted_monoid::RangeAddRangeSum<long long>;

struct RangeAddBeats : RangeAdd {
    static bool can_apply(const operator_type&, const value_type&) {
        return true;
    }
};

template <class Structure, class Update, class Get>
void check_copy_move_and_reuse(Structure base, Update update, Get get) {
    const std::size_t base_nodes = base.node_count();
    Structure current = base;
    Structure sibling = current;

    update(current, 0);
    assert(get(base) == 0);
    assert(get(sibling) == 0);
    assert(get(current) == 1);

    update(sibling, 9);
    assert(get(base) == 0);
    assert(get(current) == 1);
    assert(get(sibling) == 10);

    const std::size_t after_first_write = current.node_count();
    update(current, 1);
    assert(get(current) == 2);
    assert(current.node_count() == after_first_write);

    Structure snapshot = current;
    update(current, 2);
    assert(get(snapshot) == 2);
    assert(get(current) == 3);
    const std::size_t after_reshare = current.node_count();
    update(current, 3);
    assert(get(snapshot) == 2);
    assert(get(current) == 4);
    assert(current.node_count() == after_reshare);

    Structure moved(std::move(current));
    update(moved, 4);
    assert(get(moved) == 5);
    Structure assigned = base;
    assigned = std::move(moved);
    update(assigned, 5);
    assert(get(assigned) == 6);
    assert(get(base) == 0);

    sibling.release();
    snapshot.release();
    assigned.release();
    assert(base.node_count() == base_nodes);

    Structure reused = base;
    update(reused, 6);
    assert(get(reused) == 7);
    reused.release();
    assert(base.node_count() == base_nodes);
}

void test_segment_trees() {
    using Seg = m1une::ds::PersistentSegtree<Add>;
    check_copy_move_and_reuse(
        Seg(std::vector<long long>(8, 0)),
        [](Seg& seg, int step) { seg.set_inplace(3, step + 1); },
        [](const Seg& seg) { return seg.get(3); }
    );

    using Lazy = m1une::ds::PersistentLazySegtree<RangeAdd>;
    check_copy_move_and_reuse(
        Lazy(std::vector<long long>(8, 0)),
        [](Lazy& seg, int step) { seg.set_inplace(3, RangeAdd::make(step + 1)); },
        [](const Lazy& seg) { return seg.get(3).sum; }
    );

    Lazy base(std::vector<long long>(8, 0));
    Lazy current = base;
    current.apply_inplace(0, 8, 10);
    const std::size_t after_full_cover = current.node_count();
    current.apply_inplace(0, 8, 1);
    assert(current.node_count() == after_full_cover);
    Lazy snapshot = current;
    current.apply_inplace(2, 6, 5);
    current.set_inplace(3, RangeAdd::make(100));
    assert(base.all_prod().sum == 0);
    assert(snapshot.all_prod().sum == 88);
    assert(snapshot.get(3).sum == 11);
    assert(current.get(2).sum == 16);
    assert(current.get(3).sum == 100);

    Lazy persistent = base;
    Lazy inplace = base;
    std::uint64_t state = 1;
    for (int operation = 0; operation < 500; ++operation) {
        state = state * 6364136223846793005ULL + 1;
        int left = int(state % 9);
        state = state * 6364136223846793005ULL + 1;
        int right = int(state % 9);
        if (left > right) std::swap(left, right);
        long long add = static_cast<long long>(operation % 13) - 6;
        persistent = persistent.apply(left, right, add);
        inplace.apply_inplace(left, right, add);
        if (operation % 17 == 0) {
            int index = operation % 8;
            auto value = RangeAdd::make(operation);
            persistent = persistent.set(index, value);
            inplace.set_inplace(index, value);
        }
        auto inplace_values = inplace.to_vector();
        auto persistent_values = persistent.to_vector();
        assert(inplace_values.size() == persistent_values.size());
        for (int i = 0; i < int(inplace_values.size()); ++i) {
            assert(inplace_values[i].sum == persistent_values[i].sum);
        }
    }
    assert(base.all_prod().sum == 0);

    using Dual = m1une::ds::PersistentDualSegtree<Add>;
    Dual dual_base(std::vector<long long>(8, 0));
    Dual dual = dual_base;
    dual.apply_inplace(1, 7, 4);
    dual.apply_inplace(3, 6);
    dual.set_inplace(5, 20);
    assert(dual_base.get(5) == 0);
    assert(dual.get(3) == 10);
    assert(dual.get(5) == 20);

    using Beats = m1une::ds::PersistentSegtreeBeats<RangeAddBeats>;
    Beats beats_base(std::vector<long long>(8, 0));
    Beats beats = beats_base;
    beats.apply_inplace(1, 7, 4);
    const std::size_t beats_nodes = beats.node_count();
    beats.apply_inplace(1, 7, 3);
    assert(beats.node_count() == beats_nodes);
    Beats beats_copy = beats;
    beats.set_inplace(3, RangeAddBeats::make(50));
    assert(beats_base.all_prod().sum == 0);
    assert(beats_copy.get(3).sum == 7);
    assert(beats.get(3).sum == 50);
}

void test_dynamic_segment_trees() {
    using Seg = m1une::ds::PersistentDynamicSegtree<Add>;
    Seg base(-100, 100);
    Seg current = base;
    current.set_inplace(17, 1);
    const std::size_t nodes = current.node_count();
    current.set_inplace(17, 2);
    assert(current.node_count() == nodes);
    Seg copy = current;
    current.set_inplace(17, 3);
    assert(base.get(17) == 0);
    assert(copy.get(17) == 2);
    assert(current.get(17) == 3);

    using Lazy = m1une::ds::PersistentDynamicLazySegtree<RangeAdd>;
    Lazy lazy_base(-64, 64, RangeAdd::make(0));
    Lazy lazy = lazy_base;
    lazy.apply_inplace(-20, 30, 5);
    Lazy lazy_copy = lazy;
    lazy.apply_inplace(-5, 10, 7);
    lazy.set_inplace(0, RangeAdd::make(100));
    assert(lazy_base.get(0).sum == 0);
    assert(lazy_copy.get(0).sum == 5);
    assert(lazy.get(-6).sum == 5);
    assert(lazy.get(-5).sum == 12);
    assert(lazy.get(0).sum == 100);

    using Dual = m1une::ds::PersistentDynamicDualSegtree<Add>;
    Dual dual_base(-64, 64, 0);
    Dual dual = dual_base;
    dual.apply_inplace(-20, 30, 5);
    Dual dual_copy = dual;
    dual.apply_inplace(-5, 10, 7);
    dual.set_inplace(0, 100);
    assert(dual_base.get(0) == 0);
    assert(dual_copy.get(0) == 5);
    assert(dual.get(-5) == 12);
    assert(dual.get(0) == 100);
}

void test_dynamic_arrays() {
    using Array = m1une::ds::PersistentDynamicArray<int>;
    Array base = {0, 1, 2, 3, 4, 5};
    Array reversed = base.reverse(1, 6);
    Array copy = reversed;
    reversed.set_inplace(1, 20);
    const std::size_t nodes = reversed.node_count();
    reversed.set_inplace(1, 21);
    assert(reversed.node_count() == nodes);
    assert(base.to_vector() == (std::vector<int>{0, 1, 2, 3, 4, 5}));
    assert(copy.to_vector() == (std::vector<int>{0, 5, 4, 3, 2, 1}));
    assert(reversed.to_vector() == (std::vector<int>{0, 21, 4, 3, 2, 1}));

    using MonoidArray = m1une::ds::PersistentDynamicMonoidArray<Add>;
    MonoidArray monoid(std::vector<long long>{1, 2, 3, 4});
    MonoidArray monoid_copy = monoid;
    monoid.set_inplace(2, 30);
    assert(monoid_copy.all_prod() == 10);
    assert(monoid.all_prod() == 37);

    using LazyArray = m1une::ds::PersistentDynamicLazyMonoidArray<RangeAdd>;
    LazyArray lazy_base(std::vector<long long>{1, 2, 3, 4, 5, 6});
    LazyArray lazy = lazy_base.apply(1, 6, 10).reverse(0, 5);
    LazyArray lazy_copy = lazy;
    lazy.apply_inplace(1, 5, 7);
    lazy.set_inplace(2, RangeAdd::make(100));
    assert(lazy_base.all_prod().sum == 21);
    std::vector<long long> copy_expected = {15, 14, 13, 12, 1, 16};
    auto copy_values = lazy_copy.to_vector();
    for (int i = 0; i < int(copy_expected.size()); ++i) {
        assert(copy_values[i].sum == copy_expected[i]);
    }
    std::vector<long long> expected = {15, 21, 100, 19, 8, 16};
    auto values = lazy.to_vector();
    for (int i = 0; i < int(expected.size()); ++i) assert(values[i].sum == expected[i]);
    assert(lazy.all_prod().sum == 179);
}

void test_dsu() {
    m1une::ds::PersistentDsu base(8);
    auto current = base;
    assert(current.merge_inplace(0, 1));
    const std::size_t after_merge = current.node_count();
    assert(!current.merge_inplace(0, 1));
    assert(current.node_count() == after_merge);
    auto copy = current;
    assert(current.merge_inplace(1, 2));
    assert(!base.same(0, 1));
    assert(!copy.same(0, 2));
    assert(current.same(0, 2));

    using PotentialDsu = m1une::ds::PersistentPotentializedDsu<Add>;
    PotentialDsu potential_base(8);
    PotentialDsu potential = potential_base;
    assert(potential.merge_inplace(0, 1, 3));
    PotentialDsu potential_copy = potential;
    assert(potential.merge_inplace(1, 2, 4));
    assert(!potential.merge_inplace(0, 2, 8));
    assert(potential.diff(0, 2) == 7);
    assert(!potential_base.same(0, 1));
    assert(!potential_copy.same(0, 2));
}

void test_ordered_multiset() {
    using Multiset = m1une::ds::PersistentOrderedMultiset<int>;
    Multiset base = {1, 2, 2, 3};
    Multiset current = base;
    current.insert_inplace(2, 3);
    const std::size_t after_first_write = current.node_count();
    current.insert_inplace(2);
    assert(current.node_count() == after_first_write);
    Multiset copy = current;
    assert(current.erase_inplace(2));
    assert(base.count(2) == 2);
    assert(copy.count(2) == 6);
    assert(current.count(2) == 5);
    current.insert_inplace(4);
    assert(!base.contains(4));
    assert(current.contains(4));
    assert(current.erase_all_inplace(4));
    assert(!current.contains(4));
    assert(!current.erase_inplace(9));
}

}  // namespace

int main() {
    test_segment_trees();
    test_dynamic_segment_trees();
    test_dynamic_arrays();
    test_dsu();
    test_ordered_multiset();

    m1une::utilities::FastInput input;
    m1une::utilities::FastOutput output;
    long long a, b;
    input >> a >> b;
    output << a + b << '\n';
}
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