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:heavy_check_mark: verify/ds/dynamic_tree/lazy_link_cut_tree.test.cpp

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Code

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

#include <cassert>
#include "../../../utilities/fast_io.hpp"
#include <random>
#include <utility>
#include <vector>

#include "../../../acted_monoid/range_add_range_sum.hpp"
#include "../../../ds/dynamic_tree/lazy_link_cut_tree.hpp"

using AddSum = m1une::acted_monoid::RangeAddRangeSum<long long>;
using Node = AddSum::value_type;

void test_vertex_path_and_subtree_updates() {
    m1une::ds::LazyLinkCutTree<AddSum> lct(std::vector<long long>{1, 2, 3, 4, 5});

    assert(lct.link(0, 1));
    assert(lct.link(1, 2));
    assert(lct.link(1, 3));
    assert(lct.link(3, 4));

    Node subtree = lct.subtree_prod(0, 1);
    assert(subtree.sum == 14);
    assert(subtree.size == 4);
    assert(lct.subtree_size(0, 1) == 4);

    lct.apply(2, 4, 10);
    Node path = lct.path_prod(2, 4);
    assert(path.sum == 54);
    assert(path.size == 4);
    subtree = lct.subtree_prod(0, 1);
    assert(subtree.sum == 54);
    assert(subtree.size == 4);
    assert(lct.get(0).sum == 1);
    assert(lct.subtree_prod(1, 3).sum == 29);

    lct.apply(0, 3, 5);
    subtree = lct.subtree_prod(0, 1);
    assert(subtree.sum == 64);
    assert(subtree.size == 4);
    assert(lct.path_prod(2, 4).sum == 64);

    subtree = lct.subtree_prod(2, 1);
    assert(subtree.sum == 57);
    assert(subtree.size == 4);
}

void test_edge_path_and_subtree_updates() {
    m1une::ds::LazyLinkCutTree<AddSum> lct(3);

    int e01 = lct.link_edge(0, 1, 5);
    int e12 = lct.link_edge(1, 2, 7);
    assert(e01 == 0);
    assert(e12 == 1);

    Node full = lct.subtree_prod(0, 0);
    assert(full.sum == 12);
    assert(full.size == 2);
    Node child = lct.subtree_prod(0, 1);
    assert(child.sum == 7);
    assert(child.size == 1);
    assert(lct.subtree_size(0, 1) == 3);

    lct.apply(0, 2, 3);
    assert(lct.path_prod(0, 2).sum == 18);
    assert(lct.subtree_prod(0, 1).sum == 10);

    lct.apply(1, 2, 2);
    assert(lct.get_edge(e01).sum == 8);
    assert(lct.get_edge(e12).sum == 12);
    assert(lct.subtree_prod(0, 0).sum == 20);

    assert(lct.cut_edge(e01));
    assert(!lct.connected(0, 2));
}

void test_rooted_tree_utility_apis() {
    m1une::ds::LazyLinkCutTree<AddSum> lct(std::vector<long long>{1, 2, 3, 4, 5});

    assert(lct.link_parent(1, 0));
    assert(lct.link_parent(2, 0));
    assert(lct.link_parent(3, 2));
    assert(lct.link_parent(4, 2));

    assert(lct.root(4) == 0);
    lct.reroot(0);

    Node component = lct.component_prod(0);
    assert(component.sum == 15);
    assert(component.size == 5);
    assert(lct.component_size(0) == 5);

    assert(lct.child_toward(0, 4) == 2);
    assert(lct.child_toward(2, 4) == 4);
    assert(lct.child_toward(4, 0) == 2);

    Node branch = lct.branch_prod(0, 4);
    assert(branch.sum == 12);
    assert(branch.size == 3);
    assert(lct.branch_size(0, 4) == 3);

    assert(lct.parent(0, 4) == 2);
    assert(lct.parent(0, 2) == 0);
    assert(lct.parent(0, 0) == -1);

    Node subtree = lct.subtree_prod(0, 2);
    assert(subtree.sum == 12);
    assert(subtree.size == 3);
    assert(lct.subtree_size(0, 2) == 3);

    Node excluding = lct.subtree_prod_excluding_child(0, 2, 4);
    assert(excluding.sum == 7);
    assert(excluding.size == 2);
    assert(lct.subtree_size_excluding_child(0, 2, 4) == 2);

    lct.reroot(0);
    assert(lct.cut_parent(4));

    assert(!lct.connected(4, 0));
    component = lct.component_prod(0);
    assert(component.sum == 10);
    assert(component.size == 4);
    assert(lct.component_size(0) == 4);
    component = lct.component_prod(4);
    assert(component.sum == 5);
    assert(component.size == 1);
    assert(lct.component_size(4) == 1);

    m1une::ds::LazyLinkCutTree<AddSum> lct2(std::vector<long long>{1, 2, 3, 4, 5});
    assert(lct2.link_parent(1, 0));
    assert(lct2.link_parent(2, 0));
    assert(lct2.link_parent(3, 2));
    assert(lct2.link_parent(4, 2));
    lct2.apply(3, 4, 10);
    lct2.reroot(0);

    assert(lct2.cut_parent(2));
    assert(!lct2.connected(2, 0));
    component = lct2.component_prod(0);
    assert(component.sum == 3);
    assert(component.size == 2);
    assert(lct2.component_size(0) == 2);
    component = lct2.component_prod(2);
    assert(component.sum == 42);
    assert(component.size == 3);
    assert(lct2.component_size(2) == 3);
}

bool naive_connected(const std::vector<std::vector<int>>& adj, int s, int t) {
    std::vector<int> parent(adj.size(), -1);
    std::vector<int> stack;
    parent[s] = s;
    stack.push_back(s);
    for (int it = 0; it < int(stack.size()); it++) {
        int v = stack[it];
        if (v == t) return true;
        for (int to : adj[v]) {
            if (parent[to] != -1) continue;
            parent[to] = v;
            stack.push_back(to);
        }
    }
    return false;
}

std::vector<int> naive_path_vertices(const std::vector<std::vector<int>>& adj, int s, int t) {
    std::vector<int> parent(adj.size(), -1);
    std::vector<int> stack;
    parent[s] = s;
    stack.push_back(s);
    for (int it = 0; it < int(stack.size()); it++) {
        int v = stack[it];
        if (v == t) break;
        for (int to : adj[v]) {
            if (parent[to] != -1) continue;
            parent[to] = v;
            stack.push_back(to);
        }
    }
    assert(parent[t] != -1);
    std::vector<int> path;
    for (int v = t; v != s; v = parent[v]) path.push_back(v);
    path.push_back(s);
    return path;
}

long long naive_path_sum(const std::vector<std::vector<int>>& adj, const std::vector<long long>& value, int s, int t) {
    long long res = 0;
    for (int v : naive_path_vertices(adj, s, t)) res += value[v];
    return res;
}

long long naive_subtree_sum(const std::vector<std::vector<int>>& adj, const std::vector<long long>& value, int root,
                            int v) {
    std::vector<int> parent(adj.size(), -1);
    std::vector<int> stack;
    parent[root] = root;
    stack.push_back(root);
    for (int it = 0; it < int(stack.size()); it++) {
        int x = stack[it];
        for (int to : adj[x]) {
            if (parent[to] != -1) continue;
            parent[to] = x;
            stack.push_back(to);
        }
    }
    assert(parent[v] != -1);

    long long res = 0;
    stack.clear();
    stack.push_back(v);
    while (!stack.empty()) {
        int x = stack.back();
        stack.pop_back();
        res += value[x];
        for (int to : adj[x]) {
            if (to == parent[x]) continue;
            stack.push_back(to);
        }
    }
    return res;
}

void test_random_vertex_path_updates() {
    constexpr int n = 8;
    std::vector<long long> initial;
    std::vector<long long> value;
    for (int i = 0; i < n; i++) {
        initial.push_back(i + 1);
        value.push_back(i + 1);
    }
    m1une::ds::LazyLinkCutTree<AddSum> lct(initial);
    std::vector<std::vector<int>> adj(n);
    std::vector<std::pair<int, int>> edges;
    std::mt19937 rng(2);

    for (int step = 0; step < 700; step++) {
        int op = int(rng() % 6);
        int u = int(rng() % n);
        int v = int(rng() % n);
        if (op == 0) {
            if (u != v && !naive_connected(adj, u, v)) {
                assert(lct.link(u, v));
                adj[u].push_back(v);
                adj[v].push_back(u);
                edges.emplace_back(u, v);
            }
        } else if (op == 1) {
            if (!edges.empty()) {
                int id = int(rng() % edges.size());
                auto [a, b] = edges[id];
                assert(lct.cut(a, b));
                for (int i = 0; i < int(adj[a].size()); i++) {
                    if (adj[a][i] == b) {
                        adj[a][i] = adj[a].back();
                        adj[a].pop_back();
                        break;
                    }
                }
                for (int i = 0; i < int(adj[b].size()); i++) {
                    if (adj[b][i] == a) {
                        adj[b][i] = adj[b].back();
                        adj[b].pop_back();
                        break;
                    }
                }
                edges[id] = edges.back();
                edges.pop_back();
            }
        } else if (op == 2) {
            long long x = int(rng() % 200) - 100;
            value[u] = x;
            lct.set(u, x);
        } else if (u != v && naive_connected(adj, u, v)) {
            if (op == 3) {
                long long add = int(rng() % 21) - 10;
                lct.apply(u, v, add);
                for (int x : naive_path_vertices(adj, u, v)) value[x] += add;
            } else if (op == 4) {
                assert(lct.path_prod(u, v).sum == naive_path_sum(adj, value, u, v));
            } else {
                assert(lct.subtree_prod(u, v).sum == naive_subtree_sum(adj, value, u, v));
                assert(lct.subtree_prod(v, u).sum == naive_subtree_sum(adj, value, v, u));
            }
        }
    }
}

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    test_vertex_path_and_subtree_updates();
    test_edge_path_and_subtree_updates();
    test_rooted_tree_utility_apis();
    test_random_vertex_path_updates();

    long long a, b;
    fast_input >> a >> b;
    fast_output << a + b << '\n';
}
#line 1 "verify/ds/dynamic_tree/lazy_link_cut_tree.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"

#include <cassert>
#line 1 "utilities/fast_io.hpp"



#include <algorithm>
#include <array>
#include <cerrno>
#include <charconv>
#include <cstddef>
#include <cstdio>
#include <cstdlib>
#include <cstdint>
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#include <type_traits>
#include <utility>
#include <unistd.h>
#include <vector>

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 5 "verify/ds/dynamic_tree/lazy_link_cut_tree.test.cpp"
#include <random>
#line 8 "verify/ds/dynamic_tree/lazy_link_cut_tree.test.cpp"

#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/dynamic_tree/lazy_link_cut_tree.hpp"



#line 5 "ds/dynamic_tree/lazy_link_cut_tree.hpp"
#include <concepts>
#line 9 "ds/dynamic_tree/lazy_link_cut_tree.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 11 "ds/dynamic_tree/lazy_link_cut_tree.hpp"

namespace m1une {
namespace ds {

template <m1une::acted_monoid::IsCommutativeActedGroup ActedGroup>
struct LazyLinkCutTree {
    using T = typename ActedGroup::value_type;
    using F = typename ActedGroup::operator_type;

   private:
    struct Node {
        int left = -1;
        int right = -1;
        int parent = -1;
        bool rev = false;
        int size = 1;
        int virtual_size = 0;
        int rake_size = 0;
        int all_size = 1;
        T value = ActedGroup::id();
        T prod = ActedGroup::id();
        T rev_prod = ActedGroup::id();
        T virtual_prod = ActedGroup::id();
        T rake_prod = ActedGroup::id();
        T all_prod = ActedGroup::id();
        F lazy = ActedGroup::op_id();
    };

    struct EdgeInfo {
        int u = -1;
        int v = -1;
        int node = -1;
        bool alive = false;
    };

    std::vector<Node> _nodes;
    std::vector<EdgeInfo> _edges;
    std::vector<int> _path_buffer;

    static T make_node_value(const T& value, int) {
        return value;
    }

    static T make_node_value(T&& value, int) {
        return std::move(value);
    }

    template <class U>
    requires (!std::same_as<U, T>) && (
        requires(U x) { ActedGroup::make(x); } ||
        requires(U x, int i) { ActedGroup::make(x, i); } ||
        std::convertible_to<U, T>
    )
    static T make_node_value(const U& value, int index) {
        if constexpr (requires(U x) { ActedGroup::make(x); }) {
            return ActedGroup::make(value);
        } else if constexpr (requires(U x, int i) { ActedGroup::make(x, i); }) {
            return ActedGroup::make(value, index);
        } else {
            return static_cast<T>(value);
        }
    }

    int child_size(int node) const {
        return node == -1 ? 0 : _nodes[node].size;
    }

    int child_all_size(int node) const {
        return node == -1 ? 0 : _nodes[node].all_size;
    }

    int child_rake_size(int node) const {
        return node == -1 ? 0 : _nodes[node].rake_size;
    }

    T child_prod(int node) const {
        return node == -1 ? ActedGroup::id() : _nodes[node].prod;
    }

    T child_rev_prod(int node) const {
        return node == -1 ? ActedGroup::id() : _nodes[node].rev_prod;
    }

    T child_all_prod(int node) const {
        return node == -1 ? ActedGroup::id() : _nodes[node].all_prod;
    }

    T child_rake_prod(int node) const {
        return node == -1 ? ActedGroup::id() : _nodes[node].rake_prod;
    }

    T node_subtree_prod(int node) const {
        const Node& x = _nodes[node];
        return ActedGroup::op(x.value, x.virtual_prod);
    }

    int node_subtree_size(int node) const {
        return 1 + _nodes[node].virtual_size;
    }

    bool is_splay_root(int node) const {
        int parent = _nodes[node].parent;
        return parent == -1 || (_nodes[parent].left != node && _nodes[parent].right != node);
    }

    void update(int node) {
        Node& x = _nodes[node];
        x.size = 1 + child_size(x.left) + child_size(x.right);
        x.rake_size = x.virtual_size + child_rake_size(x.left) + child_rake_size(x.right);
        x.all_size = x.size + x.rake_size;
        x.prod = ActedGroup::op(ActedGroup::op(child_prod(x.left), x.value), child_prod(x.right));
        x.rev_prod = ActedGroup::op(ActedGroup::op(child_rev_prod(x.right), x.value), child_rev_prod(x.left));
        x.rake_prod = ActedGroup::op(ActedGroup::op(child_rake_prod(x.left), x.virtual_prod),
                                      child_rake_prod(x.right));
        x.all_prod = ActedGroup::op(x.prod, x.rake_prod);
    }

    void add_virtual_child(int node, int child) {
        if (child == -1) return;
        Node& x = _nodes[node];
        x.virtual_size += _nodes[child].all_size;
        x.virtual_prod = ActedGroup::op(x.virtual_prod, _nodes[child].all_prod);
    }

    void remove_virtual_child(int node, int child) {
        if (child == -1) return;
        Node& x = _nodes[node];
        x.virtual_size -= _nodes[child].all_size;
        x.virtual_prod = ActedGroup::op(x.virtual_prod, ActedGroup::inv(_nodes[child].all_prod));
    }

    void apply_reverse(int node) {
        if (node == -1) return;
        Node& x = _nodes[node];
        std::swap(x.left, x.right);
        std::swap(x.prod, x.rev_prod);
        x.rev = !x.rev;
    }

    void apply_operator(int node, const F& f) {
        if (node == -1) return;
        Node& x = _nodes[node];
        x.value = ActedGroup::mapping(f, x.value);
        x.prod = ActedGroup::mapping(f, x.prod);
        x.rev_prod = ActedGroup::mapping(f, x.rev_prod);
        x.all_prod = ActedGroup::op(x.prod, x.rake_prod);
        x.lazy = ActedGroup::op_comp(f, x.lazy);
    }

    void push(int node) {
        if (node == -1) return;
        Node& x = _nodes[node];
        if (x.rev) {
            apply_reverse(x.left);
            apply_reverse(x.right);
            x.rev = false;
        }
        apply_operator(x.left, x.lazy);
        apply_operator(x.right, x.lazy);
        x.lazy = ActedGroup::op_id();
    }

    void push_to(int node) {
        _path_buffer.clear();
        int cur = node;
        _path_buffer.push_back(cur);
        while (!is_splay_root(cur)) {
            cur = _nodes[cur].parent;
            _path_buffer.push_back(cur);
        }
        for (int i = int(_path_buffer.size()) - 1; i >= 0; i--) push(_path_buffer[i]);
    }

    void rotate(int node) {
        int parent = _nodes[node].parent;
        int grand = _nodes[parent].parent;
        bool is_right = _nodes[parent].right == node;
        int middle = is_right ? _nodes[node].left : _nodes[node].right;

        if (!is_splay_root(parent)) {
            if (_nodes[grand].left == parent) {
                _nodes[grand].left = node;
            } else {
                _nodes[grand].right = node;
            }
        }
        _nodes[node].parent = grand;

        if (is_right) {
            _nodes[node].left = parent;
            _nodes[parent].right = middle;
        } else {
            _nodes[node].right = parent;
            _nodes[parent].left = middle;
        }
        if (middle != -1) _nodes[middle].parent = parent;
        _nodes[parent].parent = node;

        update(parent);
        update(node);
    }

    void splay(int node) {
        push_to(node);
        while (!is_splay_root(node)) {
            int parent = _nodes[node].parent;
            int grand = _nodes[parent].parent;
            if (!is_splay_root(parent)) {
                bool zig_zig = (_nodes[parent].left == node) == (_nodes[grand].left == parent);
                rotate(zig_zig ? parent : node);
            }
            rotate(node);
        }
    }

    int access(int node) {
        int last = -1;
        for (int cur = node; cur != -1; cur = _nodes[cur].parent) {
            splay(cur);
            add_virtual_child(cur, _nodes[cur].right);
            remove_virtual_child(cur, last);
            _nodes[cur].right = last;
            if (last != -1) _nodes[last].parent = cur;
            update(cur);
            last = cur;
        }
        splay(node);
        return last;
    }

    void check_vertex(int v) const {
        assert(0 <= v && v < int(_nodes.size()));
    }

    void check_edge(int edge_id) const {
        assert(0 <= edge_id && edge_id < int(_edges.size()));
    }

   public:
    LazyLinkCutTree() = default;

    explicit LazyLinkCutTree(int n) {
        assert(0 <= n);
        _nodes.reserve(n);
        for (int i = 0; i < n; i++) add_vertex();
    }

    explicit LazyLinkCutTree(const std::vector<T>& values) {
        _nodes.reserve(values.size());
        for (int i = 0; i < int(values.size()); i++) add_vertex(values[i]);
    }

    explicit LazyLinkCutTree(std::vector<T>&& values) {
        _nodes.reserve(values.size());
        for (int i = 0; i < int(values.size()); i++) add_vertex(std::move(values[i]));
    }

    template <class U>
    requires (!std::same_as<U, T>) && (
        requires(U x) { ActedGroup::make(x); } ||
        requires(U x, int i) { ActedGroup::make(x, i); } ||
        std::convertible_to<U, T>
    )
    explicit LazyLinkCutTree(const std::vector<U>& values) {
        _nodes.reserve(values.size());
        for (int i = 0; i < int(values.size()); i++) add_vertex(make_node_value(values[i], i));
    }

    int size() const {
        return int(_nodes.size());
    }

    bool empty() const {
        return _nodes.empty();
    }

    int add_vertex(const T& value = ActedGroup::id()) {
        Node node;
        node.value = value;
        node.prod = value;
        node.rev_prod = value;
        node.all_prod = value;
        _nodes.push_back(std::move(node));
        return int(_nodes.size()) - 1;
    }

    int add_vertex(T&& value) {
        Node node;
        node.value = std::move(value);
        node.prod = node.value;
        node.rev_prod = node.value;
        node.all_prod = node.value;
        _nodes.push_back(std::move(node));
        return int(_nodes.size()) - 1;
    }

    template <class U>
    requires (!std::same_as<std::remove_cvref_t<U>, T>) && (
        requires(U x) { ActedGroup::make(x); } ||
        requires(U x, int i) { ActedGroup::make(x, i); } ||
        std::convertible_to<U, T>
    )
    int add_vertex(const U& value) {
        return add_vertex(make_node_value(value, size()));
    }

    int edge_count() const {
        return int(_edges.size());
    }

    bool edge_alive(int edge_id) const {
        check_edge(edge_id);
        return _edges[edge_id].alive;
    }

    int edge_node(int edge_id) const {
        check_edge(edge_id);
        return _edges[edge_id].node;
    }

    std::pair<int, int> edge_endpoints(int edge_id) const {
        check_edge(edge_id);
        return {_edges[edge_id].u, _edges[edge_id].v};
    }

    T get(int v) {
        check_vertex(v);
        access(v);
        return _nodes[v].value;
    }

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

    void set(int v, const T& value) {
        check_vertex(v);
        access(v);
        _nodes[v].value = value;
        update(v);
    }

    void set(int v, T&& value) {
        check_vertex(v);
        access(v);
        _nodes[v].value = std::move(value);
        update(v);
    }

    template <class U>
    requires (!std::same_as<std::remove_cvref_t<U>, T>) && (
        requires(U x) { ActedGroup::make(x); } ||
        requires(U x, int i) { ActedGroup::make(x, i); } ||
        std::convertible_to<U, T>
    )
    void set(int v, const U& value) {
        set(v, make_node_value(value, v));
    }

    // Applies `f` to one vertex.
    void apply(int v, const F& f) {
        check_vertex(v);
        access(v);
        _nodes[v].value = ActedGroup::mapping(f, _nodes[v].value);
        update(v);
    }

    // Applies `f` to the path from `u` to `v`. Internally calls `evert(u)`,
    // so the represented root may change.
    void apply(int u, int v, const F& f) {
        check_vertex(u);
        check_vertex(v);
        assert(connected(u, v));
        evert(u);
        access(v);
        apply_operator(v, f);
    }

    // Makes `v` the represented root of its component.
    void evert(int v) {
        check_vertex(v);
        access(v);
        apply_reverse(v);
    }

    // Alias for `evert(v)`; changes the represented root to `v`.
    void reroot(int v) {
        evert(v);
    }

    // Returns the current represented root of `v`'s component.
    int component_root(int v) {
        check_vertex(v);
        access(v);
        int cur = v;
        push(cur);
        while (_nodes[cur].left != -1) {
            cur = _nodes[cur].left;
            push(cur);
        }
        splay(cur);
        return cur;
    }

    // Alias for `component_root(v)`.
    int root(int v) {
        return component_root(v);
    }

    bool connected(int u, int v) {
        check_vertex(u);
        check_vertex(v);
        if (u == v) return true;
        return component_root(u) == component_root(v);
    }

    bool same(int u, int v) {
        return connected(u, v);
    }

    // Links two components. Internally calls `evert(u)`, so the represented root may change.
    bool link(int u, int v) {
        check_vertex(u);
        check_vertex(v);
        if (u == v) return false;
        evert(u);
        if (component_root(v) == u) return false;
        access(v);
        _nodes[u].parent = v;
        add_virtual_child(v, u);
        update(v);
        return true;
    }

    // Links `child` under `parent`. This is the same operation as `link(child, parent)`;
    // it internally calls `evert(child)`, so that side's represented root may change.
    bool link_parent(int child, int parent) {
        return link(child, parent);
    }

    int link_edge(int u, int v, const T& value = ActedGroup::id()) {
        check_vertex(u);
        check_vertex(v);
        if (u == v || connected(u, v)) return -1;
        int edge_id = int(_edges.size());
        int node = add_vertex(value);
        _edges.push_back(EdgeInfo{u, v, node, true});
        bool ok1 = link(u, node);
        bool ok2 = link(node, v);
        assert(ok1 && ok2);
        return edge_id;
    }

    int link_edge(int u, int v, T&& value) {
        check_vertex(u);
        check_vertex(v);
        if (u == v || connected(u, v)) return -1;
        int edge_id = int(_edges.size());
        int node = add_vertex(std::move(value));
        _edges.push_back(EdgeInfo{u, v, node, true});
        bool ok1 = link(u, node);
        bool ok2 = link(node, v);
        assert(ok1 && ok2);
        return edge_id;
    }

    template <class U>
    requires (!std::same_as<std::remove_cvref_t<U>, T>) && (
        requires(U x) { ActedGroup::make(x); } ||
        requires(U x, int i) { ActedGroup::make(x, i); } ||
        std::convertible_to<U, T>
    )
    int link_edge(int u, int v, const U& value) {
        check_vertex(u);
        check_vertex(v);
        if (u == v || connected(u, v)) return -1;
        return link_edge(u, v, make_node_value(value, size()));
    }

    // Cuts edge `(u, v)`. Internally calls `evert(u)`, so the represented root may change.
    bool cut(int u, int v) {
        check_vertex(u);
        check_vertex(v);
        if (u == v) return false;
        evert(u);
        access(v);
        if (_nodes[v].left != u || _nodes[u].right != -1) return false;
        _nodes[v].left = -1;
        _nodes[u].parent = -1;
        update(v);
        return true;
    }

    // Cuts the parent edge of `v` in the current represented-root orientation.
    // Unlike `cut(u, v)`, this does not call `evert`.
    bool cut_parent(int v) {
        check_vertex(v);
        access(v);
        int left = _nodes[v].left;
        if (left == -1) return false;
        _nodes[v].left = -1;
        _nodes[left].parent = -1;
        update(v);
        return true;
    }

    bool cut_edge(int edge_id) {
        check_edge(edge_id);
        EdgeInfo& edge = _edges[edge_id];
        if (!edge.alive) return false;
        bool ok1 = cut(edge.u, edge.node);
        bool ok2 = cut(edge.node, edge.v);
        if (ok1 && ok2) edge.alive = false;
        return ok1 && ok2;
    }

    T get_edge(int edge_id) {
        return get(edge_node(edge_id));
    }

    void set_edge(int edge_id, const T& value) {
        set(edge_node(edge_id), value);
    }

    void set_edge(int edge_id, T&& value) {
        set(edge_node(edge_id), std::move(value));
    }

    template <class U>
    requires (!std::same_as<std::remove_cvref_t<U>, T>) && (
        requires(U x) { ActedGroup::make(x); } ||
        requires(U x, int i) { ActedGroup::make(x, i); } ||
        std::convertible_to<U, T>
    )
    void set_edge(int edge_id, const U& value) {
        set(edge_node(edge_id), make_node_value(value, edge_node(edge_id)));
    }

    void apply_edge(int edge_id, const F& f) {
        apply(edge_node(edge_id), f);
    }

    // Returns the path product from `u` to `v`. Internally calls `evert(u)`,
    // so the represented root may change.
    T prod(int u, int v) {
        check_vertex(u);
        check_vertex(v);
        assert(connected(u, v));
        evert(u);
        access(v);
        return _nodes[v].prod;
    }

    // Alias for `prod(u, v)`. Internally calls `evert(u)`,
    // so the represented root may change.
    T path_prod(int u, int v) {
        return prod(u, v);
    }

    // Returns the number of vertices on path `u`-`v`. Internally calls `evert(u)`,
    // so the represented root may change.
    int path_size(int u, int v) {
        check_vertex(u);
        check_vertex(v);
        assert(connected(u, v));
        evert(u);
        access(v);
        return _nodes[v].size;
    }

    // Returns the `k`-th vertex on path `u`-`v`. Internally calls `evert(u)`,
    // so the represented root may change.
    int kth_vertex(int u, int v, int k) {
        check_vertex(u);
        check_vertex(v);
        assert(connected(u, v));
        evert(u);
        access(v);
        assert(0 <= k && k < _nodes[v].size);

        int cur = v;
        while (true) {
            push(cur);
            int left_size = child_size(_nodes[cur].left);
            if (k < left_size) {
                cur = _nodes[cur].left;
            } else if (k == left_size) {
                splay(cur);
                return cur;
            } else {
                k -= left_size + 1;
                cur = _nodes[cur].right;
            }
        }
    }

    int lca(int u, int v) {
        check_vertex(u);
        check_vertex(v);
        if (!connected(u, v)) return -1;
        if (u == v) return u;
        access(u);
        return access(v);
    }

    // Returns the aggregate of `v`'s subtree when the represented tree is rooted at `root`.
    // Internally calls `evert(root)`, so the represented root may change.
    T subtree_prod(int root, int v) {
        check_vertex(root);
        check_vertex(v);
        assert(connected(root, v));
        evert(root);
        access(v);
        return node_subtree_prod(v);
    }

    // Returns the aggregate of `v`'s subtree with respect to the current represented root.
    T subtree_prod(int v) {
        check_vertex(v);
        access(v);
        return node_subtree_prod(v);
    }

    // Returns the size of `v`'s subtree when the represented tree is rooted at `root`.
    // Internally calls `evert(root)`, so the represented root may change.
    int subtree_size(int root, int v) {
        check_vertex(root);
        check_vertex(v);
        assert(connected(root, v));
        evert(root);
        access(v);
        return node_subtree_size(v);
    }

    // Returns the size of `v`'s subtree with respect to the current represented root.
    int subtree_size(int v) {
        check_vertex(v);
        access(v);
        return node_subtree_size(v);
    }

    // Returns the aggregate of the whole connected component containing `v`.
    T component_prod(int v) {
        int r = root(v);
        return subtree_prod(r, r);
    }

    // Returns the number of vertices in the connected component containing `v`.
    int component_size(int v) {
        int r = root(v);
        return subtree_size(r, r);
    }

    // Returns the child of `root` that lies on path `root`-`v`.
    int child_toward(int root, int v) {
        check_vertex(root);
        check_vertex(v);
        assert(root != v);
        assert(connected(root, v));
        return kth_vertex(root, v, 1);
    }

    // Returns the aggregate of the entire branch of `root` that contains `v`.
    T branch_prod(int root, int v) {
        check_vertex(root);
        check_vertex(v);
        assert(root != v);
        int child = child_toward(root, v);
        return subtree_prod(root, child);
    }

    // Returns the size of the entire branch of `root` that contains `v`.
    int branch_size(int root, int v) {
        check_vertex(root);
        check_vertex(v);
        assert(root != v);
        int child = child_toward(root, v);
        return subtree_size(root, child);
    }

    // Returns the parent of `v` when rooted at `root`, or `-1` if `v == root`.
    int parent(int root, int v) {
        check_vertex(root);
        check_vertex(v);
        if (root == v) return -1;
        assert(connected(root, v));
        int d = path_size(root, v);
        assert(2 <= d);
        return kth_vertex(root, v, d - 2);
    }

    // Returns `v`'s rooted subtree aggregate excluding the child-side subtree.
    T subtree_prod_excluding_child(int root, int v, int child) {
        check_vertex(root);
        check_vertex(v);
        check_vertex(child);
        assert(parent(root, child) == v);
        T whole = subtree_prod(root, v);
        T sub = subtree_prod(root, child);
        return ActedGroup::op(whole, ActedGroup::inv(sub));
    }

    // Returns `v`'s rooted subtree size excluding the child-side subtree.
    int subtree_size_excluding_child(int root, int v, int child) {
        check_vertex(root);
        check_vertex(v);
        check_vertex(child);
        assert(parent(root, child) == v);
        return subtree_size(root, v) - subtree_size(root, child);
    }
};

}  // namespace ds
}  // namespace m1une


#line 11 "verify/ds/dynamic_tree/lazy_link_cut_tree.test.cpp"

using AddSum = m1une::acted_monoid::RangeAddRangeSum<long long>;
using Node = AddSum::value_type;

void test_vertex_path_and_subtree_updates() {
    m1une::ds::LazyLinkCutTree<AddSum> lct(std::vector<long long>{1, 2, 3, 4, 5});

    assert(lct.link(0, 1));
    assert(lct.link(1, 2));
    assert(lct.link(1, 3));
    assert(lct.link(3, 4));

    Node subtree = lct.subtree_prod(0, 1);
    assert(subtree.sum == 14);
    assert(subtree.size == 4);
    assert(lct.subtree_size(0, 1) == 4);

    lct.apply(2, 4, 10);
    Node path = lct.path_prod(2, 4);
    assert(path.sum == 54);
    assert(path.size == 4);
    subtree = lct.subtree_prod(0, 1);
    assert(subtree.sum == 54);
    assert(subtree.size == 4);
    assert(lct.get(0).sum == 1);
    assert(lct.subtree_prod(1, 3).sum == 29);

    lct.apply(0, 3, 5);
    subtree = lct.subtree_prod(0, 1);
    assert(subtree.sum == 64);
    assert(subtree.size == 4);
    assert(lct.path_prod(2, 4).sum == 64);

    subtree = lct.subtree_prod(2, 1);
    assert(subtree.sum == 57);
    assert(subtree.size == 4);
}

void test_edge_path_and_subtree_updates() {
    m1une::ds::LazyLinkCutTree<AddSum> lct(3);

    int e01 = lct.link_edge(0, 1, 5);
    int e12 = lct.link_edge(1, 2, 7);
    assert(e01 == 0);
    assert(e12 == 1);

    Node full = lct.subtree_prod(0, 0);
    assert(full.sum == 12);
    assert(full.size == 2);
    Node child = lct.subtree_prod(0, 1);
    assert(child.sum == 7);
    assert(child.size == 1);
    assert(lct.subtree_size(0, 1) == 3);

    lct.apply(0, 2, 3);
    assert(lct.path_prod(0, 2).sum == 18);
    assert(lct.subtree_prod(0, 1).sum == 10);

    lct.apply(1, 2, 2);
    assert(lct.get_edge(e01).sum == 8);
    assert(lct.get_edge(e12).sum == 12);
    assert(lct.subtree_prod(0, 0).sum == 20);

    assert(lct.cut_edge(e01));
    assert(!lct.connected(0, 2));
}

void test_rooted_tree_utility_apis() {
    m1une::ds::LazyLinkCutTree<AddSum> lct(std::vector<long long>{1, 2, 3, 4, 5});

    assert(lct.link_parent(1, 0));
    assert(lct.link_parent(2, 0));
    assert(lct.link_parent(3, 2));
    assert(lct.link_parent(4, 2));

    assert(lct.root(4) == 0);
    lct.reroot(0);

    Node component = lct.component_prod(0);
    assert(component.sum == 15);
    assert(component.size == 5);
    assert(lct.component_size(0) == 5);

    assert(lct.child_toward(0, 4) == 2);
    assert(lct.child_toward(2, 4) == 4);
    assert(lct.child_toward(4, 0) == 2);

    Node branch = lct.branch_prod(0, 4);
    assert(branch.sum == 12);
    assert(branch.size == 3);
    assert(lct.branch_size(0, 4) == 3);

    assert(lct.parent(0, 4) == 2);
    assert(lct.parent(0, 2) == 0);
    assert(lct.parent(0, 0) == -1);

    Node subtree = lct.subtree_prod(0, 2);
    assert(subtree.sum == 12);
    assert(subtree.size == 3);
    assert(lct.subtree_size(0, 2) == 3);

    Node excluding = lct.subtree_prod_excluding_child(0, 2, 4);
    assert(excluding.sum == 7);
    assert(excluding.size == 2);
    assert(lct.subtree_size_excluding_child(0, 2, 4) == 2);

    lct.reroot(0);
    assert(lct.cut_parent(4));

    assert(!lct.connected(4, 0));
    component = lct.component_prod(0);
    assert(component.sum == 10);
    assert(component.size == 4);
    assert(lct.component_size(0) == 4);
    component = lct.component_prod(4);
    assert(component.sum == 5);
    assert(component.size == 1);
    assert(lct.component_size(4) == 1);

    m1une::ds::LazyLinkCutTree<AddSum> lct2(std::vector<long long>{1, 2, 3, 4, 5});
    assert(lct2.link_parent(1, 0));
    assert(lct2.link_parent(2, 0));
    assert(lct2.link_parent(3, 2));
    assert(lct2.link_parent(4, 2));
    lct2.apply(3, 4, 10);
    lct2.reroot(0);

    assert(lct2.cut_parent(2));
    assert(!lct2.connected(2, 0));
    component = lct2.component_prod(0);
    assert(component.sum == 3);
    assert(component.size == 2);
    assert(lct2.component_size(0) == 2);
    component = lct2.component_prod(2);
    assert(component.sum == 42);
    assert(component.size == 3);
    assert(lct2.component_size(2) == 3);
}

bool naive_connected(const std::vector<std::vector<int>>& adj, int s, int t) {
    std::vector<int> parent(adj.size(), -1);
    std::vector<int> stack;
    parent[s] = s;
    stack.push_back(s);
    for (int it = 0; it < int(stack.size()); it++) {
        int v = stack[it];
        if (v == t) return true;
        for (int to : adj[v]) {
            if (parent[to] != -1) continue;
            parent[to] = v;
            stack.push_back(to);
        }
    }
    return false;
}

std::vector<int> naive_path_vertices(const std::vector<std::vector<int>>& adj, int s, int t) {
    std::vector<int> parent(adj.size(), -1);
    std::vector<int> stack;
    parent[s] = s;
    stack.push_back(s);
    for (int it = 0; it < int(stack.size()); it++) {
        int v = stack[it];
        if (v == t) break;
        for (int to : adj[v]) {
            if (parent[to] != -1) continue;
            parent[to] = v;
            stack.push_back(to);
        }
    }
    assert(parent[t] != -1);
    std::vector<int> path;
    for (int v = t; v != s; v = parent[v]) path.push_back(v);
    path.push_back(s);
    return path;
}

long long naive_path_sum(const std::vector<std::vector<int>>& adj, const std::vector<long long>& value, int s, int t) {
    long long res = 0;
    for (int v : naive_path_vertices(adj, s, t)) res += value[v];
    return res;
}

long long naive_subtree_sum(const std::vector<std::vector<int>>& adj, const std::vector<long long>& value, int root,
                            int v) {
    std::vector<int> parent(adj.size(), -1);
    std::vector<int> stack;
    parent[root] = root;
    stack.push_back(root);
    for (int it = 0; it < int(stack.size()); it++) {
        int x = stack[it];
        for (int to : adj[x]) {
            if (parent[to] != -1) continue;
            parent[to] = x;
            stack.push_back(to);
        }
    }
    assert(parent[v] != -1);

    long long res = 0;
    stack.clear();
    stack.push_back(v);
    while (!stack.empty()) {
        int x = stack.back();
        stack.pop_back();
        res += value[x];
        for (int to : adj[x]) {
            if (to == parent[x]) continue;
            stack.push_back(to);
        }
    }
    return res;
}

void test_random_vertex_path_updates() {
    constexpr int n = 8;
    std::vector<long long> initial;
    std::vector<long long> value;
    for (int i = 0; i < n; i++) {
        initial.push_back(i + 1);
        value.push_back(i + 1);
    }
    m1une::ds::LazyLinkCutTree<AddSum> lct(initial);
    std::vector<std::vector<int>> adj(n);
    std::vector<std::pair<int, int>> edges;
    std::mt19937 rng(2);

    for (int step = 0; step < 700; step++) {
        int op = int(rng() % 6);
        int u = int(rng() % n);
        int v = int(rng() % n);
        if (op == 0) {
            if (u != v && !naive_connected(adj, u, v)) {
                assert(lct.link(u, v));
                adj[u].push_back(v);
                adj[v].push_back(u);
                edges.emplace_back(u, v);
            }
        } else if (op == 1) {
            if (!edges.empty()) {
                int id = int(rng() % edges.size());
                auto [a, b] = edges[id];
                assert(lct.cut(a, b));
                for (int i = 0; i < int(adj[a].size()); i++) {
                    if (adj[a][i] == b) {
                        adj[a][i] = adj[a].back();
                        adj[a].pop_back();
                        break;
                    }
                }
                for (int i = 0; i < int(adj[b].size()); i++) {
                    if (adj[b][i] == a) {
                        adj[b][i] = adj[b].back();
                        adj[b].pop_back();
                        break;
                    }
                }
                edges[id] = edges.back();
                edges.pop_back();
            }
        } else if (op == 2) {
            long long x = int(rng() % 200) - 100;
            value[u] = x;
            lct.set(u, x);
        } else if (u != v && naive_connected(adj, u, v)) {
            if (op == 3) {
                long long add = int(rng() % 21) - 10;
                lct.apply(u, v, add);
                for (int x : naive_path_vertices(adj, u, v)) value[x] += add;
            } else if (op == 4) {
                assert(lct.path_prod(u, v).sum == naive_path_sum(adj, value, u, v));
            } else {
                assert(lct.subtree_prod(u, v).sum == naive_subtree_sum(adj, value, u, v));
                assert(lct.subtree_prod(v, u).sum == naive_subtree_sum(adj, value, v, u));
            }
        }
    }
}

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    test_vertex_path_and_subtree_updates();
    test_edge_path_and_subtree_updates();
    test_rooted_tree_utility_apis();
    test_random_vertex_path_updates();

    long long a, b;
    fast_input >> a >> b;
    fast_output << a + b << '\n';
}
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