verify/ds/segtree/persistent_segtree_beats.test.cpp

• View this file on GitHub
• Last update: 2026-10-06 01:57:24+09:00
• Problem: https://judge.yosupo.jp/problem/persistent_range_affine_range_sum

Depends on

• Acted Monoid Concept (acted_monoid/concept.hpp)
• Range Affine Range Sum (acted_monoid/range_affine_range_sum.hpp)
• Range AP Add Range Sum (acted_monoid/range_ap_add_range_sum.hpp)
• Beats Acted Monoid Concept (beats_acted_monoid/concept.hpp)
• ds/segtree/persistent_node_pool.hpp
• ds/segtree/persistent_segtree_beats.hpp
• ModInt (math/modint.hpp)
• Fast IO (utilities/fast_io.hpp)

Code

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

#include "../../../ds/segtree/persistent_segtree_beats.hpp"

#include <algorithm>
#include <cassert>
#include <cstdint>
#include <limits>
#include <numeric>
#include <optional>
#include <utility>
#include <vector>

#include "../../../acted_monoid/range_affine_range_sum.hpp"
#include "../../../acted_monoid/range_ap_add_range_sum.hpp"
#include "../../../math/modint.hpp"
#include "../../../utilities/fast_io.hpp"

namespace {

struct ChminRangeSum {
    struct value_type {
        long long sum;
        long long maximum;
        long long second_maximum;
        int maximum_count;
        int length;
    };

    using operator_type = long long;

    static constexpr long long negative_infinity =
        std::numeric_limits<long long>::lowest();
    static constexpr long long positive_infinity =
        std::numeric_limits<long long>::max();

    static value_type id() {
        return {0, negative_infinity, negative_infinity, 0, 0};
    }

    static value_type op(const value_type& left, const value_type& right) {
        value_type result;
        result.sum = left.sum + right.sum;
        result.length = left.length + right.length;
        if (left.maximum == right.maximum) {
            result.maximum = left.maximum;
            result.second_maximum = std::max(
                left.second_maximum,
                right.second_maximum
            );
            result.maximum_count =
                left.maximum_count + right.maximum_count;
        } else if (left.maximum < right.maximum) {
            result.maximum = right.maximum;
            result.second_maximum = std::max(
                left.maximum,
                right.second_maximum
            );
            result.maximum_count = right.maximum_count;
        } else {
            result.maximum = left.maximum;
            result.second_maximum = std::max(
                left.second_maximum,
                right.maximum
            );
            result.maximum_count = left.maximum_count;
        }
        return result;
    }

    static operator_type op_id() {
        return positive_infinity;
    }

    static operator_type op_comp(operator_type f, operator_type g) {
        return std::min(f, g);
    }

    static bool can_apply(operator_type f, const value_type& value) {
        return value.maximum <= f || value.second_maximum < f;
    }

    static value_type mapping(operator_type f, value_type value) {
        if (value.maximum <= f) return value;
        assert(value.second_maximum < f);
        value.sum +=
            (f - value.maximum) * static_cast<long long>(
                value.maximum_count
            );
        value.maximum = f;
        return value;
    }

    static value_type make(long long value) {
        return {value, value, negative_infinity, 1, 1};
    }
};

struct RangeApAddRangeSumBeats
    : m1une::acted_monoid::RangeApAddRangeSum<long long> {
    static bool can_apply(
        const operator_type&,
        const value_type&
    ) {
        return true;
    }
};

void check_version(
    const m1une::ds::PersistentSegtreeBeats<ChminRangeSum>& seg,
    const std::vector<long long>& expected
) {
    assert(seg.size() == int(expected.size()));
    assert(seg.empty() == expected.empty());
    assert(
        seg.all_prod().sum
        == std::accumulate(expected.begin(), expected.end(), 0LL)
    );

    std::vector<ChminRangeSum::value_type> materialized =
        seg.to_vector();
    assert(materialized.size() == expected.size());
    for (int index = 0; index < int(expected.size()); ++index) {
        assert(materialized[index].sum == expected[index]);
        assert(seg.get(index).sum == expected[index]);
        assert(seg[index].sum == expected[index]);
    }

    int slice_left = int(expected.size()) / 4;
    int slice_right = int(expected.size()) * 3 / 4;
    std::vector<ChminRangeSum::value_type> slice =
        seg.to_vector(slice_left, slice_right);
    assert(int(slice.size()) == slice_right - slice_left);
    for (int index = slice_left; index < slice_right; ++index) {
        assert(slice[index - slice_left].sum == expected[index]);
    }

    for (int left = 0; left <= int(expected.size()); ++left) {
        long long sum = 0;
        for (int right = left; right <= int(expected.size()); ++right) {
            assert(seg.prod(left, right).sum == sum);
            if (right < int(expected.size())) sum += expected[right];
        }
    }

    if (!expected.empty()) {
        int left = int(expected.size()) / 3;
        long long limit = 0;
        int expected_right = left;
        while (
            expected_right < int(expected.size()) &&
            limit + expected[expected_right] <= 60
        ) {
            limit += expected[expected_right++];
        }
        assert(seg.max_right(
            left,
            [](const ChminRangeSum::value_type& value) {
                return value.sum <= 60;
            }
        ) == expected_right);

        int right = int(expected.size()) * 2 / 3 + 1;
        right = std::min(right, int(expected.size()));
        limit = 0;
        int expected_left = right;
        while (
            expected_left > 0 &&
            limit + expected[expected_left - 1] <= 60
        ) {
            limit += expected[--expected_left];
        }
        assert(seg.min_left(
            right,
            [](const ChminRangeSum::value_type& value) {
                return value.sum <= 60;
            }
        ) == expected_left);
    }
}

void test_randomized_persistence() {
    using Seg = m1une::ds::PersistentSegtreeBeats<ChminRangeSum>;

    Seg empty;
    assert(empty.empty());
    assert(empty.size() == 0);
    assert(empty.all_prod().sum == 0);
    assert(empty.prod(0, 0).sum == 0);
    assert(empty.to_vector().empty());
    assert(empty.max_right(0, [](const ChminRangeSum::value_type&) {
        return true;
    }) == 0);
    assert(empty.min_left(0, [](const ChminRangeSum::value_type&) {
        return true;
    }) == 0);

    std::uint64_t state = 0x6a09e667f3bcc909ULL;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int trial = 0; trial < 80; ++trial) {
        int size = int(random() % 20) + 1;
        std::vector<long long> initial(size);
        for (long long& value : initial) {
            value = static_cast<long long>(random() % 51);
        }

        std::vector<std::optional<Seg>> versions;
        std::vector<std::vector<long long>> naive_versions;
        versions.emplace_back(std::in_place, initial);
        naive_versions.push_back(initial);

        for (int operation = 0; operation < 240; ++operation) {
            int base = int(random() % versions.size());
            int left = int(random() % (size + 1));
            int right = int(random() % (size + 1));
            if (right < left) std::swap(left, right);
            int type = int(random() % 5);

            std::vector<long long> next = naive_versions[base];
            if (type <= 1) {
                long long upper = static_cast<long long>(random() % 51);
                versions.push_back(
                    versions[base]->apply(left, right, upper)
                );
                for (int index = left; index < right; ++index) {
                    next[index] = std::min(next[index], upper);
                }
                naive_versions.push_back(std::move(next));
            } else if (type == 2) {
                int index = int(random() % size);
                long long value = static_cast<long long>(random() % 51);
                versions.push_back(versions[base]->set(
                    index,
                    ChminRangeSum::make(value)
                ));
                next[index] = value;
                naive_versions.push_back(std::move(next));
            } else if (type == 3) {
                int source = int(random() % versions.size());
                versions.push_back(versions[base]->copy_range_from(
                    *versions[source],
                    left,
                    right
                ));
                std::copy(
                    naive_versions[source].begin() + left,
                    naive_versions[source].begin() + right,
                    next.begin() + left
                );
                naive_versions.push_back(std::move(next));
            } else {
                long long expected = std::accumulate(
                    next.begin() + left,
                    next.begin() + right,
                    0LL
                );
                assert(versions[base]->prod(left, right).sum == expected);
            }

            int checked = int(random() % versions.size());
            check_version(*versions[checked], naive_versions[checked]);
        }
        check_version(*versions[0], initial);

        Seg base(initial);
        std::size_t before = base.node_count();
        {
            Seg disposable = base.apply(0, size, 17);
            assert(base.node_count() >= before);
            check_version(base, initial);
        }
        assert(base.node_count() == before);

        Seg released = base;
        released.release();
        assert(released.empty());
        assert(base.node_count() == before);
        check_version(base, initial);
    }
}

void test_index_aware_action() {
    using Seg =
        m1une::ds::PersistentSegtreeBeats<RangeApAddRangeSumBeats>;
    Seg original(std::vector<long long>{1, 2, 3, 4, 5});
    Seg updated = original.apply(1, 5, std::pair<long long, long long>(2, 3));
    std::vector<long long> expected{1, 5, 8, 11, 14};
    assert(original.all_prod().sum == 15);
    assert(updated.all_prod().sum == 39);
    for (int index = 0; index < 5; ++index) {
        assert(updated[index].sum == expected[index]);
    }
}

}  // namespace

using mint = m1une::math::modint998244353;

struct RangeAffineRangeSumBeats
    : m1une::acted_monoid::RangeAffineRangeSum<mint> {
    static bool can_apply(
        const operator_type&,
        const value_type&
    ) {
        return true;
    }
};

int main() {
    test_randomized_persistence();
    test_index_aware_action();

    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    int size, query_count;
    fast_input >> size >> query_count;
    std::vector<mint> initial(size);
    for (mint& value : initial) fast_input >> value;

    using Seg =
        m1une::ds::PersistentSegtreeBeats<RangeAffineRangeSumBeats>;
    std::vector<std::optional<Seg>> versions(query_count + 1);
    versions[0].emplace(initial);
    for (int query = 0; query < query_count; ++query) {
        int type, version, left, right;
        fast_input >> type >> version;
        ++version;
        if (type == 0) {
            mint multiplier, addition;
            fast_input >> left >> right >> multiplier >> addition;
            versions[query + 1] = versions[version]->apply(
                left,
                right,
                std::pair<mint, mint>(multiplier, addition)
            );
        } else if (type == 1) {
            int source;
            fast_input >> source >> left >> right;
            ++source;
            versions[query + 1] = versions[version]->copy_range_from(
                *versions[source],
                left,
                right
            );
        } else {
            fast_input >> left >> right;
            fast_output << versions[version]->prod(left, right).sum << '\n';
        }
    }
}

#line 1 "verify/ds/segtree/persistent_segtree_beats.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/persistent_range_affine_range_sum"

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



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

#line 1 "beats_acted_monoid/concept.hpp"



#line 5 "beats_acted_monoid/concept.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 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 1 "ds/segtree/persistent_node_pool.hpp"



#line 6 "ds/segtree/persistent_node_pool.hpp"
#include <limits>
#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 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 4 "verify/ds/segtree/persistent_segtree_beats.test.cpp"

#include <algorithm>
#line 7 "verify/ds/segtree/persistent_segtree_beats.test.cpp"
#include <cstdint>
#line 9 "verify/ds/segtree/persistent_segtree_beats.test.cpp"
#include <numeric>
#include <optional>
#line 13 "verify/ds/segtree/persistent_segtree_beats.test.cpp"

#line 1 "acted_monoid/range_affine_range_sum.hpp"



#line 5 "acted_monoid/range_affine_range_sum.hpp"

namespace m1une {
namespace acted_monoid {

template <typename T>
struct RangeAffineRangeSumNode {
    T sum;
    int size;
};

// Designed to accept Modint or similar types as T
template <typename T>
struct RangeAffineRangeSum {
    using value_type = RangeAffineRangeSumNode<T>;
    using operator_type = std::pair<T, T>;  // {a, b} for ax + b
    static constexpr bool commutative = true;
    static constexpr bool operator_commutative = false;

    // Value Monoid
    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 int size(const value_type& value) {
        return value.size;
    }

    // Operator Monoid (Affine Composition)
    // f(x) = a1*x + b1, g(x) = a2*x + b2
    // f(g(x)) = a1*(a2*x + b2) + b1 = (a1*a2)*x + (a1*b2 + b1)
    static constexpr operator_type op_id() {
        return {T(1), T(0)};
    }
    static constexpr operator_type op_comp(const operator_type& f, const operator_type& g) {
        return {f.first * g.first, f.first * g.second + f.second};
    }

    // Mapping
    // \sum (a*x_i + b) = a * \sum x_i + b * size
    static constexpr value_type mapping(const operator_type& f, const value_type& x) {
        return {f.first * x.sum + f.second * T(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 "acted_monoid/range_ap_add_range_sum.hpp"



#line 5 "acted_monoid/range_ap_add_range_sum.hpp"

namespace m1une {
namespace acted_monoid {

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

template <typename T>
struct RangeApAddRangeSum {
    using value_type = RangeApAddRangeSumNode<T>;
    using operator_type = std::pair<T, T>;  // {a, b} for adding a * i + b
    static constexpr bool commutative = false;
    static constexpr bool operator_commutative = true;

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

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

    static constexpr value_type mapping(const operator_type& f, const value_type& x) {
        return mapping(f, x, 0);
    }

    static constexpr value_type mapping(const operator_type& f, const value_type& x, long long ord) {
        return {x.sum + f.first * (x.ord_sum + T(ord) * T(x.size)) + f.second * T(x.size), x.size, x.ord_sum};
    }

    static constexpr operator_type op_shift(const operator_type& f, long long ord) {
        return {f.first, f.second + f.first * T(ord)};
    }

    static constexpr operator_type op_reverse(const operator_type& f, long long size) {
        return {-f.first, f.second + f.first * T(size - 1)};
    }

    static constexpr value_type make(const T& val) {
        return {val, 1, T(0)};
    }
};

}  // namespace acted_monoid
}  // namespace m1une


#line 1 "math/modint.hpp"



#line 6 "math/modint.hpp"
#include <iostream>
#include <type_traits>
#line 9 "math/modint.hpp"

namespace m1une {
namespace math {

template <uint32_t Modulus>
struct ModInt {
    static_assert(0 < Modulus, "Modulus must be positive");

   private:
    uint32_t _v;

   public:
    static constexpr uint32_t mod() {
        return Modulus;
    }

    static constexpr ModInt raw(uint32_t v) noexcept {
        ModInt x;
        x._v = v;
        return x;
    }

    constexpr ModInt() noexcept : _v(0) {}

    template <class Integer, std::enable_if_t<std::is_integral_v<Integer>, int> = 0>
    constexpr ModInt(Integer v) noexcept {
        if constexpr (std::is_signed_v<Integer>) {
            int64_t x = static_cast<int64_t>(v) % static_cast<int64_t>(Modulus);
            if (x < 0) x += Modulus;
            _v = static_cast<uint32_t>(x);
        } else {
            _v = static_cast<uint32_t>(static_cast<uint64_t>(v) % Modulus);
        }
    }

    constexpr uint32_t val() const noexcept {
        return _v;
    }

    constexpr ModInt& operator++() noexcept {
        _v++;
        if (_v == Modulus) _v = 0;
        return *this;
    }

    constexpr ModInt& operator--() noexcept {
        if (_v == 0) _v = Modulus;
        _v--;
        return *this;
    }

    constexpr ModInt operator++(int) noexcept {
        ModInt res = *this;
        ++*this;
        return res;
    }

    constexpr ModInt operator--(int) noexcept {
        ModInt res = *this;
        --*this;
        return res;
    }

    constexpr ModInt& operator+=(const ModInt& rhs) noexcept {
        _v += rhs._v;
        if (_v >= Modulus) _v -= Modulus;
        return *this;
    }

    constexpr ModInt& operator-=(const ModInt& rhs) noexcept {
        _v -= rhs._v;
        if (_v >= Modulus) _v += Modulus;
        return *this;
    }

    constexpr ModInt& operator*=(const ModInt& rhs) noexcept {
        uint64_t z = _v;
        z *= rhs._v;
        _v = static_cast<uint32_t>(z % Modulus);
        return *this;
    }

    constexpr ModInt& operator/=(const ModInt& rhs) noexcept {
        return *this *= rhs.inv();
    }

    constexpr ModInt operator+(const ModInt& rhs) const noexcept {
        return ModInt(*this) += rhs;
    }
    constexpr ModInt operator-(const ModInt& rhs) const noexcept {
        return ModInt(*this) -= rhs;
    }
    constexpr ModInt operator*(const ModInt& rhs) const noexcept {
        return ModInt(*this) *= rhs;
    }
    constexpr ModInt operator/(const ModInt& rhs) const noexcept {
        return ModInt(*this) /= rhs;
    }

    constexpr bool operator==(const ModInt& rhs) const noexcept {
        return _v == rhs._v;
    }
    constexpr bool operator!=(const ModInt& rhs) const noexcept {
        return _v != rhs._v;
    }

    constexpr ModInt pow(long long n) const noexcept {
        ModInt res = raw(1 % Modulus);
        ModInt x = n < 0 ? inv() : *this;
        uint64_t exponent = n < 0 ? uint64_t(-(n + 1)) + 1 : uint64_t(n);
        while (exponent > 0) {
            if (exponent & 1) res *= x;
            x *= x;
            exponent >>= 1;
        }
        return res;
    }

    constexpr ModInt inv() const noexcept {
        int64_t a = _v, b = Modulus, u = 1, v = 0;
        while (b) {
            int64_t t = a / b;
            a -= t * b;
            std::swap(a, b);
            u -= t * v;
            std::swap(u, v);
        }
        assert(a == 1);
        u %= Modulus;
        if (u < 0) u += Modulus;
        return raw(static_cast<uint32_t>(u));
    }

    friend std::ostream& operator<<(std::ostream& os, const ModInt& rhs) {
        return os << rhs._v;
    }

    friend std::istream& operator>>(std::istream& is, ModInt& rhs) {
        long long v;
        is >> v;
        rhs = ModInt(v);
        return is;
    }
};

using modint998244353 = ModInt<998244353>;
using modint1000000007 = ModInt<1000000007>;

template <int Id = 0>
struct DynamicModInt {
   private:
    uint32_t _v;
    inline static uint32_t _mod = 1;

   public:
    static uint32_t mod() noexcept {
        return _mod;
    }

    static void set_mod(uint32_t modulus) noexcept {
        assert(modulus > 0);
        assert(modulus <= uint32_t(1) << 31);
        _mod = modulus;
    }

    static DynamicModInt raw(uint32_t v) noexcept {
        assert(v < _mod);
        DynamicModInt x;
        x._v = v;
        return x;
    }

    DynamicModInt() noexcept : _v(0) {}

    template <class Integer, std::enable_if_t<std::is_integral_v<Integer>, int> = 0>
    DynamicModInt(Integer v) noexcept {
        if constexpr (std::is_signed_v<Integer>) {
            int64_t x = static_cast<int64_t>(v) % static_cast<int64_t>(_mod);
            if (x < 0) x += _mod;
            _v = static_cast<uint32_t>(x);
        } else {
            _v = static_cast<uint32_t>(static_cast<uint64_t>(v) % _mod);
        }
    }

    uint32_t val() const noexcept {
        return _v;
    }

    DynamicModInt& operator++() noexcept {
        _v++;
        if (_v == _mod) _v = 0;
        return *this;
    }

    DynamicModInt& operator--() noexcept {
        if (_v == 0) _v = _mod;
        _v--;
        return *this;
    }

    DynamicModInt operator++(int) noexcept {
        DynamicModInt result = *this;
        ++*this;
        return result;
    }

    DynamicModInt operator--(int) noexcept {
        DynamicModInt result = *this;
        --*this;
        return result;
    }

    DynamicModInt& operator+=(const DynamicModInt& rhs) noexcept {
        _v += rhs._v;
        if (_v >= _mod) _v -= _mod;
        return *this;
    }

    DynamicModInt& operator-=(const DynamicModInt& rhs) noexcept {
        _v -= rhs._v;
        if (_v >= _mod) _v += _mod;
        return *this;
    }

    DynamicModInt& operator*=(const DynamicModInt& rhs) noexcept {
        _v = static_cast<uint32_t>(uint64_t(_v) * rhs._v % _mod);
        return *this;
    }

    DynamicModInt& operator/=(const DynamicModInt& rhs) noexcept {
        return *this *= rhs.inv();
    }

    DynamicModInt operator+(const DynamicModInt& rhs) const noexcept {
        return DynamicModInt(*this) += rhs;
    }

    DynamicModInt operator-(const DynamicModInt& rhs) const noexcept {
        return DynamicModInt(*this) -= rhs;
    }

    DynamicModInt operator*(const DynamicModInt& rhs) const noexcept {
        return DynamicModInt(*this) *= rhs;
    }

    DynamicModInt operator/(const DynamicModInt& rhs) const noexcept {
        return DynamicModInt(*this) /= rhs;
    }

    bool operator==(const DynamicModInt& rhs) const noexcept {
        return _v == rhs._v;
    }

    bool operator!=(const DynamicModInt& rhs) const noexcept {
        return _v != rhs._v;
    }

    DynamicModInt pow(long long exponent) const noexcept {
        DynamicModInt result = raw(1 % _mod);
        DynamicModInt base = exponent < 0 ? inv() : *this;
        uint64_t magnitude =
            exponent < 0 ? uint64_t(-(exponent + 1)) + 1 : uint64_t(exponent);
        while (magnitude > 0) {
            if (magnitude & 1) result *= base;
            base *= base;
            magnitude >>= 1;
        }
        return result;
    }

    DynamicModInt inv() const noexcept {
        int64_t a = _v, b = _mod, u = 1, v = 0;
        while (b) {
            int64_t quotient = a / b;
            a -= quotient * b;
            std::swap(a, b);
            u -= quotient * v;
            std::swap(u, v);
        }
        assert(a == 1);
        u %= _mod;
        if (u < 0) u += _mod;
        return raw(static_cast<uint32_t>(u));
    }

    friend std::ostream& operator<<(std::ostream& os, const DynamicModInt& rhs) {
        return os << rhs._v;
    }

    friend std::istream& operator>>(std::istream& is, DynamicModInt& rhs) {
        long long value;
        is >> value;
        rhs = DynamicModInt(value);
        return is;
    }
};

}  // namespace math
}  // 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 18 "verify/ds/segtree/persistent_segtree_beats.test.cpp"

namespace {

struct ChminRangeSum {
    struct value_type {
        long long sum;
        long long maximum;
        long long second_maximum;
        int maximum_count;
        int length;
    };

    using operator_type = long long;

    static constexpr long long negative_infinity =
        std::numeric_limits<long long>::lowest();
    static constexpr long long positive_infinity =
        std::numeric_limits<long long>::max();

    static value_type id() {
        return {0, negative_infinity, negative_infinity, 0, 0};
    }

    static value_type op(const value_type& left, const value_type& right) {
        value_type result;
        result.sum = left.sum + right.sum;
        result.length = left.length + right.length;
        if (left.maximum == right.maximum) {
            result.maximum = left.maximum;
            result.second_maximum = std::max(
                left.second_maximum,
                right.second_maximum
            );
            result.maximum_count =
                left.maximum_count + right.maximum_count;
        } else if (left.maximum < right.maximum) {
            result.maximum = right.maximum;
            result.second_maximum = std::max(
                left.maximum,
                right.second_maximum
            );
            result.maximum_count = right.maximum_count;
        } else {
            result.maximum = left.maximum;
            result.second_maximum = std::max(
                left.second_maximum,
                right.maximum
            );
            result.maximum_count = left.maximum_count;
        }
        return result;
    }

    static operator_type op_id() {
        return positive_infinity;
    }

    static operator_type op_comp(operator_type f, operator_type g) {
        return std::min(f, g);
    }

    static bool can_apply(operator_type f, const value_type& value) {
        return value.maximum <= f || value.second_maximum < f;
    }

    static value_type mapping(operator_type f, value_type value) {
        if (value.maximum <= f) return value;
        assert(value.second_maximum < f);
        value.sum +=
            (f - value.maximum) * static_cast<long long>(
                value.maximum_count
            );
        value.maximum = f;
        return value;
    }

    static value_type make(long long value) {
        return {value, value, negative_infinity, 1, 1};
    }
};

struct RangeApAddRangeSumBeats
    : m1une::acted_monoid::RangeApAddRangeSum<long long> {
    static bool can_apply(
        const operator_type&,
        const value_type&
    ) {
        return true;
    }
};

void check_version(
    const m1une::ds::PersistentSegtreeBeats<ChminRangeSum>& seg,
    const std::vector<long long>& expected
) {
    assert(seg.size() == int(expected.size()));
    assert(seg.empty() == expected.empty());
    assert(
        seg.all_prod().sum
        == std::accumulate(expected.begin(), expected.end(), 0LL)
    );

    std::vector<ChminRangeSum::value_type> materialized =
        seg.to_vector();
    assert(materialized.size() == expected.size());
    for (int index = 0; index < int(expected.size()); ++index) {
        assert(materialized[index].sum == expected[index]);
        assert(seg.get(index).sum == expected[index]);
        assert(seg[index].sum == expected[index]);
    }

    int slice_left = int(expected.size()) / 4;
    int slice_right = int(expected.size()) * 3 / 4;
    std::vector<ChminRangeSum::value_type> slice =
        seg.to_vector(slice_left, slice_right);
    assert(int(slice.size()) == slice_right - slice_left);
    for (int index = slice_left; index < slice_right; ++index) {
        assert(slice[index - slice_left].sum == expected[index]);
    }

    for (int left = 0; left <= int(expected.size()); ++left) {
        long long sum = 0;
        for (int right = left; right <= int(expected.size()); ++right) {
            assert(seg.prod(left, right).sum == sum);
            if (right < int(expected.size())) sum += expected[right];
        }
    }

    if (!expected.empty()) {
        int left = int(expected.size()) / 3;
        long long limit = 0;
        int expected_right = left;
        while (
            expected_right < int(expected.size()) &&
            limit + expected[expected_right] <= 60
        ) {
            limit += expected[expected_right++];
        }
        assert(seg.max_right(
            left,
            [](const ChminRangeSum::value_type& value) {
                return value.sum <= 60;
            }
        ) == expected_right);

        int right = int(expected.size()) * 2 / 3 + 1;
        right = std::min(right, int(expected.size()));
        limit = 0;
        int expected_left = right;
        while (
            expected_left > 0 &&
            limit + expected[expected_left - 1] <= 60
        ) {
            limit += expected[--expected_left];
        }
        assert(seg.min_left(
            right,
            [](const ChminRangeSum::value_type& value) {
                return value.sum <= 60;
            }
        ) == expected_left);
    }
}

void test_randomized_persistence() {
    using Seg = m1une::ds::PersistentSegtreeBeats<ChminRangeSum>;

    Seg empty;
    assert(empty.empty());
    assert(empty.size() == 0);
    assert(empty.all_prod().sum == 0);
    assert(empty.prod(0, 0).sum == 0);
    assert(empty.to_vector().empty());
    assert(empty.max_right(0, [](const ChminRangeSum::value_type&) {
        return true;
    }) == 0);
    assert(empty.min_left(0, [](const ChminRangeSum::value_type&) {
        return true;
    }) == 0);

    std::uint64_t state = 0x6a09e667f3bcc909ULL;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int trial = 0; trial < 80; ++trial) {
        int size = int(random() % 20) + 1;
        std::vector<long long> initial(size);
        for (long long& value : initial) {
            value = static_cast<long long>(random() % 51);
        }

        std::vector<std::optional<Seg>> versions;
        std::vector<std::vector<long long>> naive_versions;
        versions.emplace_back(std::in_place, initial);
        naive_versions.push_back(initial);

        for (int operation = 0; operation < 240; ++operation) {
            int base = int(random() % versions.size());
            int left = int(random() % (size + 1));
            int right = int(random() % (size + 1));
            if (right < left) std::swap(left, right);
            int type = int(random() % 5);

            std::vector<long long> next = naive_versions[base];
            if (type <= 1) {
                long long upper = static_cast<long long>(random() % 51);
                versions.push_back(
                    versions[base]->apply(left, right, upper)
                );
                for (int index = left; index < right; ++index) {
                    next[index] = std::min(next[index], upper);
                }
                naive_versions.push_back(std::move(next));
            } else if (type == 2) {
                int index = int(random() % size);
                long long value = static_cast<long long>(random() % 51);
                versions.push_back(versions[base]->set(
                    index,
                    ChminRangeSum::make(value)
                ));
                next[index] = value;
                naive_versions.push_back(std::move(next));
            } else if (type == 3) {
                int source = int(random() % versions.size());
                versions.push_back(versions[base]->copy_range_from(
                    *versions[source],
                    left,
                    right
                ));
                std::copy(
                    naive_versions[source].begin() + left,
                    naive_versions[source].begin() + right,
                    next.begin() + left
                );
                naive_versions.push_back(std::move(next));
            } else {
                long long expected = std::accumulate(
                    next.begin() + left,
                    next.begin() + right,
                    0LL
                );
                assert(versions[base]->prod(left, right).sum == expected);
            }

            int checked = int(random() % versions.size());
            check_version(*versions[checked], naive_versions[checked]);
        }
        check_version(*versions[0], initial);

        Seg base(initial);
        std::size_t before = base.node_count();
        {
            Seg disposable = base.apply(0, size, 17);
            assert(base.node_count() >= before);
            check_version(base, initial);
        }
        assert(base.node_count() == before);

        Seg released = base;
        released.release();
        assert(released.empty());
        assert(base.node_count() == before);
        check_version(base, initial);
    }
}

void test_index_aware_action() {
    using Seg =
        m1une::ds::PersistentSegtreeBeats<RangeApAddRangeSumBeats>;
    Seg original(std::vector<long long>{1, 2, 3, 4, 5});
    Seg updated = original.apply(1, 5, std::pair<long long, long long>(2, 3));
    std::vector<long long> expected{1, 5, 8, 11, 14};
    assert(original.all_prod().sum == 15);
    assert(updated.all_prod().sum == 39);
    for (int index = 0; index < 5; ++index) {
        assert(updated[index].sum == expected[index]);
    }
}

}  // namespace

using mint = m1une::math::modint998244353;

struct RangeAffineRangeSumBeats
    : m1une::acted_monoid::RangeAffineRangeSum<mint> {
    static bool can_apply(
        const operator_type&,
        const value_type&
    ) {
        return true;
    }
};

int main() {
    test_randomized_persistence();
    test_index_aware_action();

    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    int size, query_count;
    fast_input >> size >> query_count;
    std::vector<mint> initial(size);
    for (mint& value : initial) fast_input >> value;

    using Seg =
        m1une::ds::PersistentSegtreeBeats<RangeAffineRangeSumBeats>;
    std::vector<std::optional<Seg>> versions(query_count + 1);
    versions[0].emplace(initial);
    for (int query = 0; query < query_count; ++query) {
        int type, version, left, right;
        fast_input >> type >> version;
        ++version;
        if (type == 0) {
            mint multiplier, addition;
            fast_input >> left >> right >> multiplier >> addition;
            versions[query + 1] = versions[version]->apply(
                left,
                right,
                std::pair<mint, mint>(multiplier, addition)
            );
        } else if (type == 1) {
            int source;
            fast_input >> source >> left >> right;
            ++source;
            versions[query + 1] = versions[version]->copy_range_from(
                *versions[source],
                left,
                right
            );
        } else {
            fast_input >> left >> right;
            fast_output << versions[version]->prod(left, right).sum << '\n';
        }
    }
}

Back to top page