verify/ds/wavelet_matrix/dynamic_wavelet_matrix_sum.test.cpp

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• Last update: 2026-10-06 01:57:24+09:00
• Problem: https://judge.yosupo.jp/problem/point_add_rectangle_sum

Depends on

• Dynamic Wavelet Matrix (ds/wavelet_matrix/dynamic_wavelet_matrix.hpp)
• Dynamic Wavelet Matrix with Sums (ds/wavelet_matrix/dynamic_wavelet_matrix_sum.hpp)
• Fast IO (utilities/fast_io.hpp)

Code

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

#include "../../../ds/wavelet_matrix/dynamic_wavelet_matrix_sum.hpp"
#include "../../../utilities/fast_io.hpp"

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

namespace {

using Matrix = m1une::ds::DynamicWaveletMatrixSum<int, long long>;

std::uint64_t random_state = 0x6a09e667f3bcc909ULL;

std::uint64_t random_value() {
    random_state ^= random_state << 7;
    random_state ^= random_state >> 9;
    return random_state;
}

int random_int(int lower, int upper) {
    return lower + int(random_value() % std::uint64_t(upper - lower + 1));
}

void check_queries(
    const Matrix& matrix,
    const std::vector<int>& values,
    const std::vector<long long>& weights
) {
    assert(values.size() == weights.size());
    assert(matrix.size() == int(values.size()));
    assert(matrix.empty() == values.empty());
    for (int i = 0; i < int(values.size()); i++) {
        assert(matrix.access(i) == values[i]);
        assert(matrix[i] == values[i]);
        assert(matrix.weight(i) == weights[i]);
        std::pair<int, long long> expected(values[i], weights[i]);
        assert(matrix.get(i) == expected);
    }

    int left = int(random_value() % (values.size() + 1));
    int right = int(random_value() % (values.size() + 1));
    if (right < left) std::swap(left, right);
    int lower = random_int(-25, 25);
    int upper = random_int(-25, 25);
    if (upper < lower) std::swap(lower, upper);

    long long total = 0;
    long long below = 0;
    long long between = 0;
    int frequency = 0;
    std::vector<std::pair<int, int>> order;
    for (int i = left; i < right; i++) {
        total += weights[i];
        if (values[i] < upper) below += weights[i];
        if (lower <= values[i] && values[i] < upper) {
            between += weights[i];
            frequency++;
        }
        order.emplace_back(values[i], i);
    }
    std::stable_sort(
        order.begin(),
        order.end(),
        [](const auto& first, const auto& second) {
            return first.first < second.first;
        }
    );

    assert(matrix.range_sum(left, right) == total);
    assert(matrix.range_sum(left, right, upper) == below);
    assert(matrix.range_sum(left, right, lower, upper) == between);
    assert(matrix.range_freq(left, right, lower, upper) == frequency);

    long long smallest_sum = 0;
    long long largest_sum = 0;
    for (int k = 0; k <= int(order.size()); k++) {
        assert(matrix.sum_k_smallest(left, right, k) == smallest_sum);
        assert(matrix.sum_k_largest(left, right, k) == largest_sum);
        if (k < int(order.size())) {
            smallest_sum += weights[order[k].second];
            largest_sum += weights[order[order.size() - 1 - k].second];
        }
    }

    if (!order.empty()) {
        int k = int(random_value() % order.size());
        assert(matrix.kth_smallest(left, right, k) == order[k].first);
        assert(
            matrix.kth_largest(left, right, k) ==
            order[order.size() - 1 - k].first
        );
    }

    int value = random_int(-20, 20);
    assert(
        matrix.rank(value, left, right) ==
        int(std::count(values.begin() + left, values.begin() + right, value))
    );
    assert(
        matrix.rank(value, right) ==
        int(std::count(values.begin(), values.begin() + right, value))
    );

    std::optional<int> previous;
    std::optional<int> next;
    for (int i = left; i < right; i++) {
        if (values[i] < upper &&
            (!previous.has_value() || previous.value() < values[i])) {
            previous = values[i];
        }
        if (lower <= values[i] &&
            (!next.has_value() || values[i] < next.value())) {
            next = values[i];
        }
    }
    assert(matrix.prev_value(left, right, upper) == previous);
    assert(matrix.next_value(left, right, lower) == next);
}

void test_randomized() {
    std::vector<int> default_values;
    default_values.push_back(std::numeric_limits<int>::min());
    default_values.push_back(-2);
    default_values.push_back(0);
    default_values.push_back(std::numeric_limits<int>::max());
    m1une::ds::DynamicWaveletMatrixSum<int, long long> default_matrix(
        default_values
    );
    assert(
        default_matrix.range_sum(0, 4) ==
        static_cast<long long>(std::numeric_limits<int>::min()) - 2 +
            std::numeric_limits<int>::max()
    );
    default_matrix.set(1, 10);
    assert(default_matrix.weight(1) == 10);
    default_matrix.insert(2, -7);
    assert(default_matrix.weight(2) == -7);

    std::vector<unsigned long long> unsigned_values;
    unsigned_values.push_back(0);
    unsigned_values.push_back(std::numeric_limits<unsigned long long>::max());
    std::vector<long long> unsigned_weights;
    unsigned_weights.push_back(3);
    unsigned_weights.push_back(4);
    m1une::ds::DynamicWaveletMatrixSum<unsigned long long, long long>
        unsigned_matrix(unsigned_values, unsigned_weights);
    assert(unsigned_matrix.kth_largest(0, 2, 0) == unsigned_values[1]);
    assert(unsigned_matrix.range_sum(0, 2) == 7);
    unsigned_matrix.clear();
    assert(unsigned_matrix.empty());

    for (int trial = 0; trial < 80; trial++) {
        int initial_size = random_int(0, 60);
        std::vector<int> values(initial_size);
        std::vector<long long> weights(initial_size);
        for (int i = 0; i < initial_size; i++) {
            values[i] = random_int(-20, 20);
            weights[i] = random_int(-50, 50);
        }
        Matrix matrix(values, weights);

        for (int operation = 0; operation < 220; operation++) {
            int type = int(random_value() % 8);
            if (values.empty()) type = 0;
            if (type == 0) {
                int position = int(random_value() % (values.size() + 1));
                int value = random_int(-20, 20);
                long long weight = random_int(-50, 50);
                values.insert(values.begin() + position, value);
                weights.insert(weights.begin() + position, weight);
                matrix.insert(position, value, weight);
            } else if (type == 1) {
                int position = int(random_value() % values.size());
                std::pair<int, long long> expected(
                    values[position],
                    weights[position]
                );
                values.erase(values.begin() + position);
                weights.erase(weights.begin() + position);
                assert(matrix.erase(position) == expected);
            } else if (type == 2) {
                int position = int(random_value() % values.size());
                int value = random_int(-20, 20);
                long long weight = random_int(-50, 50);
                values[position] = value;
                weights[position] = weight;
                matrix.set(position, value, weight);
            } else if (type == 3) {
                int position = int(random_value() % values.size());
                int value = random_int(-20, 20);
                values[position] = value;
                matrix.set_value(position, value);
            } else if (type == 4) {
                int position = int(random_value() % values.size());
                long long weight = random_int(-50, 50);
                weights[position] = weight;
                matrix.set_weight(position, weight);
            } else if (type == 5) {
                int position = int(random_value() % values.size());
                long long delta = random_int(-20, 20);
                weights[position] += delta;
                matrix.add_weight(position, delta);
            } else if (type == 6) {
                int value = random_int(-20, 20);
                values.push_back(value);
                weights.push_back(value);
                matrix.push_back(value);
            } else {
                int position = int(random_value() % values.size());
                int value = random_int(-20, 20);
                values[position] = value;
                weights[position] = value;
                matrix.set(position, value);
            }
            check_queries(matrix, values, weights);
        }
    }

    std::vector<int> values(700);
    std::vector<long long> weights(700);
    for (int i = 0; i < int(values.size()); i++) {
        values[i] = random_int(-100, 100);
        weights[i] = random_int(-1000, 1000);
    }
    Matrix matrix(values, weights);
    for (int operation = 0; operation < 2500; operation++) {
        int from = int(random_value() % values.size());
        int to = int(random_value() % values.size());
        int value = values[from];
        long long weight = weights[from];
        values.erase(values.begin() + from);
        weights.erase(weights.begin() + from);
        std::pair<int, long long> expected(value, weight);
        assert(matrix.erase(from) == expected);
        values.insert(values.begin() + to, value);
        weights.insert(weights.begin() + to, weight);
        matrix.insert(to, value, weight);
        if (operation % 25 == 0) check_queries(matrix, values, weights);
    }
}

void test_boundary_search() {
    {
        std::vector<int> equal_values(700, 7);
        std::vector<long long> equal_weights(700);
        for (int i = 0; i < int(equal_weights.size()); i++) {
            equal_weights[i] = i % 17 + 1;
        }
        Matrix equal_matrix(equal_values, equal_weights);
        for (int operation = 0; operation < 300; operation++) {
            int type = int(random_value() % 3);
            if (type == 0) {
                int position =
                    int(random_value() % (equal_values.size() + 1));
                long long weight = random_int(1, 20);
                equal_values.insert(equal_values.begin() + position, 7);
                equal_weights.insert(
                    equal_weights.begin() + position,
                    weight
                );
                equal_matrix.insert(position, 7, weight);
            } else if (type == 1 && equal_values.size() > 300) {
                int position = int(random_value() % equal_values.size());
                equal_values.erase(equal_values.begin() + position);
                equal_weights.erase(equal_weights.begin() + position);
                equal_matrix.erase(position);
            } else {
                int position = int(random_value() % equal_values.size());
                equal_weights[position] = random_int(1, 20);
                equal_matrix.set_weight(position, equal_weights[position]);
            }

            int left = int(random_value() % (equal_values.size() + 1));
            int right = int(random_value() % (equal_values.size() + 1));
            if (right < left) std::swap(left, right);
            long long total = 0;
            for (int i = left; i < right; i++) total += equal_weights[i];
            long long limit = static_cast<long long>(
                random_value() % std::uint64_t(total + 21)
            );
            auto predicate = [limit](long long sum) {
                return sum <= limit;
            };

            long long sum = 0;
            int smallest = 0;
            while (left + smallest < right &&
                   sum + equal_weights[left + smallest] <= limit) {
                sum += equal_weights[left + smallest];
                smallest++;
            }
            assert(
                equal_matrix.max_count_smallest(
                    left,
                    right,
                    predicate
                ) == smallest
            );

            sum = 0;
            int largest = 0;
            while (left + largest < right &&
                   sum + equal_weights[right - 1 - largest] <= limit) {
                sum += equal_weights[right - 1 - largest];
                largest++;
            }
            assert(
                equal_matrix.max_count_largest(left, right, predicate) ==
                largest
            );
        }
    }

    std::vector<int> values(700);
    std::vector<long long> weights(700);
    for (int i = 0; i < int(values.size()); i++) {
        values[i] = random_int(-30, 30);
        weights[i] = random_int(1, 20);
    }
    Matrix matrix(values, weights);

    for (int operation = 0; operation < 700; operation++) {
        int type = int(random_value() % 4);
        if (type == 0) {
            int position = int(random_value() % (values.size() + 1));
            int value = random_int(-30, 30);
            long long weight = random_int(1, 20);
            values.insert(values.begin() + position, value);
            weights.insert(weights.begin() + position, weight);
            matrix.insert(position, value, weight);
        } else if (type == 1 && values.size() > 300) {
            int position = int(random_value() % values.size());
            values.erase(values.begin() + position);
            weights.erase(weights.begin() + position);
            matrix.erase(position);
        } else if (type == 2) {
            int position = int(random_value() % values.size());
            values[position] = random_int(-30, 30);
            matrix.set_value(position, values[position]);
        } else {
            int position = int(random_value() % values.size());
            weights[position] = random_int(1, 20);
            matrix.set_weight(position, weights[position]);
        }

        int left = int(random_value() % (values.size() + 1));
        int right = int(random_value() % (values.size() + 1));
        if (right < left) std::swap(left, right);
        std::vector<std::pair<int, int>> order;
        long long total = 0;
        for (int i = left; i < right; i++) {
            order.emplace_back(values[i], i);
            total += weights[i];
        }
        std::stable_sort(
            order.begin(),
            order.end(),
            [](const auto& first, const auto& second) {
                return first.first < second.first;
            }
        );
        long long limit = static_cast<long long>(
            random_value() % std::uint64_t(total + 21)
        );
        auto predicate = [limit](long long sum) { return sum <= limit; };

        long long sum = 0;
        int smallest = 0;
        while (smallest < int(order.size()) &&
               sum + weights[order[smallest].second] <= limit) {
            sum += weights[order[smallest].second];
            smallest++;
        }
        assert(
            matrix.max_count_smallest(left, right, predicate) == smallest
        );

        sum = 0;
        int largest = 0;
        while (largest < int(order.size()) &&
               sum + weights[order[order.size() - 1 - largest].second] <=
                   limit) {
            sum += weights[order[order.size() - 1 - largest].second];
            largest++;
        }
        assert(matrix.max_count_largest(left, right, predicate) == largest);
    }
}

struct Point {
    int x = 0;
    int y = 0;
    long long weight = 0;
    int query_index = -1;
};

struct Query {
    int type = 0;
    int first = 0;
    int second = 0;
    int third = 0;
    int fourth = 0;
    long long weight = 0;
    int position = -1;
};

}  // namespace

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

    test_randomized();
    test_boundary_search();

    int point_count = 0;
    int query_count = 0;
    fast_input >> point_count >> query_count;
    std::vector<Point> points(point_count);
    for (Point& point : points) {
        fast_input >> point.x >> point.y >> point.weight;
    }

    std::vector<Query> queries(query_count);
    for (int query_index = 0; query_index < query_count; query_index++) {
        Query& query = queries[query_index];
        fast_input >> query.type;
        if (query.type == 0) {
            fast_input >> query.first >> query.second >> query.weight;
            Point point;
            point.x = query.first;
            point.y = query.second;
            point.query_index = query_index;
            points.push_back(point);
        } else {
            fast_input >> query.first >> query.second >> query.third >>
                query.fourth;
        }
    }

    std::stable_sort(
        points.begin(),
        points.end(),
        [](const Point& first, const Point& second) {
            return first.x < second.x;
        }
    );
    std::vector<int> xs(points.size());
    std::vector<int> ys(points.size());
    std::vector<long long> weights(points.size());
    for (int i = 0; i < int(points.size()); i++) {
        xs[i] = points[i].x;
        ys[i] = points[i].y;
        weights[i] = points[i].weight;
        if (points[i].query_index != -1) {
            queries[points[i].query_index].position = i;
        }
    }

    Matrix matrix(ys, weights);
    for (const Query& query : queries) {
        if (query.type == 0) {
            matrix.set_weight(query.position, query.weight);
        } else {
            int left = int(std::lower_bound(
                               xs.begin(),
                               xs.end(),
                               query.first
                           ) -
                           xs.begin());
            int right = int(std::lower_bound(
                                xs.begin(),
                                xs.end(),
                                query.third
                            ) -
                            xs.begin());
            fast_output << matrix.range_sum(
                               left,
                               right,
                               query.second,
                               query.fourth
                           )
                        << '\n';
        }
    }
}

#line 1 "verify/ds/wavelet_matrix/dynamic_wavelet_matrix_sum.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/point_add_rectangle_sum"

#line 1 "ds/wavelet_matrix/dynamic_wavelet_matrix_sum.hpp"



#include <algorithm>
#include <array>
#include <bit>
#include <cassert>
#include <concepts>
#include <cstdint>
#include <limits>
#include <optional>
#include <type_traits>
#include <utility>
#include <vector>

#line 1 "ds/wavelet_matrix/dynamic_wavelet_matrix.hpp"



#line 15 "ds/wavelet_matrix/dynamic_wavelet_matrix.hpp"

namespace m1une {
namespace ds {

namespace dynamic_wavelet_matrix_detail {

// A dynamic bit vector stored as an implicit treap of small packed chunks.
class DynamicRankBitVector {
   public:
    struct AccessRankResult {
        bool value;
        int ones_before;
    };

    struct EraseRankResult {
        bool value;
        int ones_before;
    };

    struct RankPair {
        int left_ones;
        int right_ones;
    };

   private:
    static constexpr int word_bits = 64;
    static constexpr int chunk_words = 4;
    static constexpr int chunk_capacity = word_bits * chunk_words;
    static constexpr int minimum_chunk_size = chunk_capacity / 2;

    struct Node {
        std::array<std::uint64_t, chunk_words> bits{};
        std::uint32_t priority = 0;
        int left = 0;
        int right = 0;
        int subtree_size = 0;
        int subtree_ones = 0;
        std::uint16_t length = 0;
        std::uint16_t chunk_ones = 0;
    };

    std::vector<Node> _nodes;
    std::vector<int> _free_nodes;
    int _root = 0;
    std::uint32_t _random_state = 1;

    int size_of(int node) const {
        return _nodes[node].subtree_size;
    }

    int ones_of(int node) const {
        return _nodes[node].subtree_ones;
    }

    std::uint32_t next_priority() {
        _random_state ^= _random_state << 13;
        _random_state ^= _random_state >> 17;
        _random_state ^= _random_state << 5;
        if (_random_state == 0) _random_state = 1;
        return _random_state;
    }

    void update(int node) {
        if (node == 0) return;
        _nodes[node].subtree_size =
            size_of(_nodes[node].left) + int(_nodes[node].length) +
            size_of(_nodes[node].right);
        _nodes[node].subtree_ones =
            ones_of(_nodes[node].left) + int(_nodes[node].chunk_ones) +
            ones_of(_nodes[node].right);
    }

    bool local_get(int node, int position) const {
        return (_nodes[node].bits[position / word_bits] >>
                (position % word_bits)) &
               1U;
    }

    void local_set(int node, int position, bool value) {
        std::uint64_t mask =
            std::uint64_t(1) << (position % word_bits);
        std::uint64_t& word = _nodes[node].bits[position / word_bits];
        if (value) {
            word |= mask;
        } else {
            word &= ~mask;
        }
    }

    int local_rank1(int node, int right) const {
        int full_words = right / word_bits;
        int result = 0;
        for (int word = 0; word < full_words; word++) {
            result += std::popcount(_nodes[node].bits[word]);
        }
        int remainder = right % word_bits;
        if (remainder != 0) {
            result += std::popcount(
                _nodes[node].bits[full_words] &
                ((std::uint64_t(1) << remainder) - 1)
            );
        }
        return result;
    }

    void clear_unused_bits(int node) {
        int length = _nodes[node].length;
        int word = length / word_bits;
        int remainder = length % word_bits;
        if (word < chunk_words) {
            if (remainder == 0) {
                _nodes[node].bits[word] = 0;
            } else {
                _nodes[node].bits[word] &=
                    (std::uint64_t(1) << remainder) - 1;
                word++;
            }
            for (; word < chunk_words; word++) {
                _nodes[node].bits[word] = 0;
            }
        }
    }

    void local_insert(int node, int position, bool value) {
        Node& current = _nodes[node];
        assert(0 <= position && position <= current.length);
        assert(current.length < chunk_capacity);

        int first_word = position / word_bits;
        int offset = position % word_bits;
        for (int word = chunk_words - 1; word > first_word; word--) {
            current.bits[word] =
                (current.bits[word] << 1) |
                (current.bits[word - 1] >> (word_bits - 1));
        }
        std::uint64_t lower_mask =
            offset == 0 ? 0 : (std::uint64_t(1) << offset) - 1;
        current.bits[first_word] =
            (current.bits[first_word] & lower_mask) |
            ((current.bits[first_word] & ~lower_mask) << 1);
        if (value) {
            current.bits[first_word] |= std::uint64_t(1) << offset;
        }
        current.length++;
        current.chunk_ones += value;
        clear_unused_bits(node);
        update(node);
    }

    bool local_erase(int node, int position) {
        Node& current = _nodes[node];
        assert(0 <= position && position < current.length);
        bool value = local_get(node, position);
        int first_word = position / word_bits;
        int offset = position % word_bits;
        std::uint64_t lower_mask =
            offset == 0 ? 0 : (std::uint64_t(1) << offset) - 1;
        std::uint64_t shifted =
            offset == word_bits - 1
                ? 0
                : (current.bits[first_word] >> (offset + 1)) << offset;
        std::uint64_t carry =
            first_word + 1 < chunk_words
                ? (current.bits[first_word + 1] & 1U) << (word_bits - 1)
                : 0;
        current.bits[first_word] =
            (current.bits[first_word] & lower_mask) | shifted | carry;
        for (int word = first_word + 1; word < chunk_words; word++) {
            std::uint64_t next_carry =
                word + 1 < chunk_words
                    ? (current.bits[word + 1] & 1U) << (word_bits - 1)
                    : 0;
            current.bits[word] = (current.bits[word] >> 1) | next_carry;
        }
        current.length--;
        current.chunk_ones -= value;
        clear_unused_bits(node);
        update(node);
        return value;
    }

    int new_node() {
        int node;
        if (_free_nodes.empty()) {
            node = int(_nodes.size());
            _nodes.emplace_back();
        } else {
            node = _free_nodes.back();
            _free_nodes.pop_back();
            _nodes[node] = Node();
        }
        _nodes[node].priority = next_priority();
        return node;
    }

    int new_node(const std::vector<std::uint8_t>& bits, int first, int last) {
        int node = new_node();
        _nodes[node].length = std::uint16_t(last - first);
        for (int position = first; position < last; position++) {
            if (bits[position]) {
                local_set(node, position - first, true);
                _nodes[node].chunk_ones++;
            }
        }
        update(node);
        return node;
    }

    void recycle_node(int node) {
        assert(node != 0);
        _nodes[node] = Node();
        _free_nodes.push_back(node);
    }

    int merge(int left, int right) {
        if (left == 0 || right == 0) return left != 0 ? left : right;
        if (_nodes[left].priority > _nodes[right].priority) {
            _nodes[left].right = merge(_nodes[left].right, right);
            update(left);
            return left;
        }
        _nodes[right].left = merge(left, _nodes[right].left);
        update(right);
        return right;
    }

    int rotate_right(int tree) {
        int result = _nodes[tree].left;
        _nodes[tree].left = _nodes[result].right;
        _nodes[result].right = tree;
        update(tree);
        update(result);
        return result;
    }

    int rotate_left(int tree) {
        int result = _nodes[tree].right;
        _nodes[tree].right = _nodes[result].left;
        _nodes[result].left = tree;
        update(tree);
        update(result);
        return result;
    }

    int extract_leftmost(int tree, int& extracted) {
        if (_nodes[tree].left == 0) {
            extracted = tree;
            int result = _nodes[tree].right;
            _nodes[tree].right = 0;
            update(tree);
            return result;
        }
        _nodes[tree].left = extract_leftmost(_nodes[tree].left, extracted);
        update(tree);
        return tree;
    }

    int extract_rightmost(int tree, int& extracted) {
        if (_nodes[tree].right == 0) {
            extracted = tree;
            int result = _nodes[tree].left;
            _nodes[tree].left = 0;
            update(tree);
            return result;
        }
        _nodes[tree].right = extract_rightmost(_nodes[tree].right, extracted);
        update(tree);
        return tree;
    }

    void assign_concatenation(
        int destination,
        int first,
        int second,
        int begin,
        int length
    ) {
        std::array<std::uint64_t, chunk_words> bits{};
        int ones = 0;
        int first_length = _nodes[first].length;
        for (int position = 0; position < length; position++) {
            int source_position = begin + position;
            bool value = source_position < first_length
                             ? local_get(first, source_position)
                             : local_get(
                                   second,
                                   source_position - first_length
                               );
            if (value) {
                bits[position / word_bits] |=
                    std::uint64_t(1) << (position % word_bits);
                ones++;
            }
        }
        _nodes[destination].bits = bits;
        _nodes[destination].length = std::uint16_t(length);
        _nodes[destination].chunk_ones = std::uint16_t(ones);
        update(destination);
    }

    int rebalance(int node) {
        if (_nodes[node].length >= minimum_chunk_size) return node;

        if (_nodes[node].right != 0) {
            int neighbor = 0;
            _nodes[node].right =
                extract_leftmost(_nodes[node].right, neighbor);
            int total = _nodes[node].length + _nodes[neighbor].length;
            if (total <= chunk_capacity) {
                assign_concatenation(node, node, neighbor, 0, total);
                recycle_node(neighbor);
            } else {
                int left_length = total / 2;
                std::array<std::uint8_t, chunk_capacity * 2> values{};
                for (int i = 0; i < _nodes[node].length; i++) {
                    values[i] = local_get(node, i);
                }
                int old_length = _nodes[node].length;
                for (int i = 0; i < _nodes[neighbor].length; i++) {
                    values[old_length + i] = local_get(neighbor, i);
                }
                std::vector<std::uint8_t> packed(values.begin(), values.begin() + total);
                assign_from_values(node, packed, 0, left_length);
                assign_from_values(neighbor, packed, left_length, total);
                _nodes[node].right = merge(neighbor, _nodes[node].right);
            }
            update(node);
            return node;
        }

        if (_nodes[node].left != 0) {
            int neighbor = 0;
            _nodes[node].left =
                extract_rightmost(_nodes[node].left, neighbor);
            int total = _nodes[neighbor].length + _nodes[node].length;
            std::array<std::uint8_t, chunk_capacity * 2> values{};
            for (int i = 0; i < _nodes[neighbor].length; i++) {
                values[i] = local_get(neighbor, i);
            }
            int neighbor_length = _nodes[neighbor].length;
            for (int i = 0; i < _nodes[node].length; i++) {
                values[neighbor_length + i] = local_get(node, i);
            }
            std::vector<std::uint8_t> packed(values.begin(), values.begin() + total);
            if (total <= chunk_capacity) {
                assign_from_values(node, packed, 0, total);
                recycle_node(neighbor);
            } else {
                int left_length = total / 2;
                assign_from_values(neighbor, packed, 0, left_length);
                assign_from_values(node, packed, left_length, total);
                _nodes[node].left = merge(_nodes[node].left, neighbor);
            }
            update(node);
        }
        return node;
    }

    void assign_from_values(
        int node,
        const std::vector<std::uint8_t>& values,
        int first,
        int last
    ) {
        _nodes[node].bits.fill(0);
        _nodes[node].length = std::uint16_t(last - first);
        _nodes[node].chunk_ones = 0;
        for (int i = first; i < last; i++) {
            if (values[i]) {
                local_set(node, i - first, true);
                _nodes[node].chunk_ones++;
            }
        }
        update(node);
    }

    int prefix_rank1_impl(int tree, int right) const {
        int result = 0;
        while (tree != 0 && right != 0) {
            int left_size = size_of(_nodes[tree].left);
            if (right <= left_size) {
                tree = _nodes[tree].left;
                continue;
            }
            result += ones_of(_nodes[tree].left);
            right -= left_size;
            int take = std::min(right, int(_nodes[tree].length));
            result += local_rank1(tree, take);
            right -= take;
            if (right == 0) break;
            tree = _nodes[tree].right;
        }
        return result;
    }

    RankPair rank1_pair_impl(int tree, int left, int right) const {
        if (left == right) {
            int ones = prefix_rank1_impl(tree, left);
            return RankPair{ones, ones};
        }
        if (tree == 0 || right == 0) return RankPair{0, 0};

        int left_size = size_of(_nodes[tree].left);
        int chunk_end = left_size + _nodes[tree].length;
        if (right <= left_size) {
            return rank1_pair_impl(_nodes[tree].left, left, right);
        }
        if (chunk_end <= left) {
            int base =
                ones_of(_nodes[tree].left) + _nodes[tree].chunk_ones;
            RankPair result = rank1_pair_impl(
                _nodes[tree].right,
                left - chunk_end,
                right - chunk_end
            );
            result.left_ones += base;
            result.right_ones += base;
            return result;
        }

        int left_ones;
        if (left <= left_size) {
            left_ones = prefix_rank1_impl(_nodes[tree].left, left);
        } else {
            left_ones = ones_of(_nodes[tree].left) +
                        local_rank1(tree, left - left_size);
        }

        int right_ones;
        if (right <= chunk_end) {
            right_ones = ones_of(_nodes[tree].left) +
                         local_rank1(tree, right - left_size);
        } else {
            right_ones = ones_of(_nodes[tree].left) +
                         _nodes[tree].chunk_ones +
                         prefix_rank1_impl(
                             _nodes[tree].right,
                             right - chunk_end
                         );
        }
        return RankPair{left_ones, right_ones};
    }

    int insert_impl(
        int tree,
        int position,
        bool value,
        int& ones_before
    ) {
        if (tree == 0) {
            int node = new_node();
            local_insert(node, 0, value);
            return node;
        }

        int left_size = size_of(_nodes[tree].left);
        int length = _nodes[tree].length;
        if (position < left_size) {
            _nodes[tree].left = insert_impl(
                _nodes[tree].left,
                position,
                value,
                ones_before
            );
            update(tree);
            if (_nodes[_nodes[tree].left].priority >
                _nodes[tree].priority) {
                tree = rotate_right(tree);
            }
            return tree;
        }
        if (position > left_size + length) {
            ones_before +=
                ones_of(_nodes[tree].left) + _nodes[tree].chunk_ones;
            _nodes[tree].right = insert_impl(
                _nodes[tree].right,
                position - left_size - length,
                value,
                ones_before
            );
            update(tree);
            if (_nodes[_nodes[tree].right].priority >
                _nodes[tree].priority) {
                tree = rotate_left(tree);
            }
            return tree;
        }

        int local_position = position - left_size;
        ones_before += ones_of(_nodes[tree].left) +
                       local_rank1(tree, local_position);
        if (length < chunk_capacity) {
            local_insert(tree, local_position, value);
            return tree;
        }

        std::vector<std::uint8_t> values(chunk_capacity);
        for (int i = 0; i < chunk_capacity; i++) {
            values[i] = local_get(tree, i);
        }
        int right_chunk = new_node();
        int middle = chunk_capacity / 2;
        assign_from_values(tree, values, 0, middle);
        assign_from_values(right_chunk, values, middle, chunk_capacity);
        if (local_position <= middle) {
            local_insert(tree, local_position, value);
        } else {
            local_insert(right_chunk, local_position - middle, value);
        }

        int old_right = _nodes[tree].right;
        _nodes[tree].right = 0;
        update(tree);
        return merge(merge(tree, right_chunk), old_right);
    }

    int erase_impl(
        int tree,
        int position,
        EraseRankResult& result
    ) {
        int left_size = size_of(_nodes[tree].left);
        int length = _nodes[tree].length;
        if (position < left_size) {
            _nodes[tree].left = erase_impl(
                _nodes[tree].left,
                position,
                result
            );
            update(tree);
            return tree;
        }
        if (position >= left_size + length) {
            result.ones_before +=
                ones_of(_nodes[tree].left) + _nodes[tree].chunk_ones;
            _nodes[tree].right = erase_impl(
                _nodes[tree].right,
                position - left_size - length,
                result
            );
            update(tree);
            return tree;
        }

        int local_position = position - left_size;
        result.ones_before += ones_of(_nodes[tree].left) +
                              local_rank1(tree, local_position);
        result.value = local_erase(tree, local_position);
        if (_nodes[tree].length == 0) {
            int merged = merge(_nodes[tree].left, _nodes[tree].right);
            recycle_node(tree);
            return merged;
        }
        return rebalance(tree);
    }

    void update_subtree(int tree) {
        if (tree == 0) return;
        update_subtree(_nodes[tree].left);
        update_subtree(_nodes[tree].right);
        update(tree);
    }

    void build(const std::vector<std::uint8_t>& bits) {
        _nodes.clear();
        _nodes.emplace_back();
        _free_nodes.clear();
        _root = 0;
        _nodes.reserve(bits.size() / minimum_chunk_size + 2);

        std::vector<int> stack;
        for (int first = 0; first < int(bits.size()); first += chunk_capacity) {
            int last = std::min(first + chunk_capacity, int(bits.size()));
            int node = new_node(bits, first, last);
            int left = 0;
            while (!stack.empty() &&
                   _nodes[stack.back()].priority < _nodes[node].priority) {
                left = stack.back();
                stack.pop_back();
            }
            _nodes[node].left = left;
            if (!stack.empty()) _nodes[stack.back()].right = node;
            stack.push_back(node);
        }
        if (!stack.empty()) _root = stack.front();
        update_subtree(_root);
    }

   public:
    DynamicRankBitVector() : _nodes(1) {}

    explicit DynamicRankBitVector(
        const std::vector<std::uint8_t>& bits,
        std::uint32_t seed = 1
    ) : _random_state(seed == 0 ? 1 : seed) {
        build(bits);
    }

    int size() const {
        return size_of(_root);
    }

    bool get(int position) const {
        return access_with_rank(position).value;
    }

    AccessRankResult access_with_rank(int position) const {
        assert(0 <= position && position < size());
        int ones_before = 0;
        int tree = _root;
        while (tree != 0) {
            int left_size = size_of(_nodes[tree].left);
            if (position < left_size) {
                tree = _nodes[tree].left;
            } else if (position < left_size + _nodes[tree].length) {
                int local_position = position - left_size;
                ones_before += ones_of(_nodes[tree].left) +
                               local_rank1(tree, local_position);
                return AccessRankResult{
                    local_get(tree, local_position),
                    ones_before
                };
            } else {
                ones_before +=
                    ones_of(_nodes[tree].left) + _nodes[tree].chunk_ones;
                position -= left_size + _nodes[tree].length;
                tree = _nodes[tree].right;
            }
        }
        assert(false);
        return AccessRankResult{false, 0};
    }

    int rank1(int right) const {
        assert(0 <= right && right <= size());
        return prefix_rank1_impl(_root, right);
    }

    RankPair rank1_pair(int left, int right) const {
        assert(0 <= left && left <= right && right <= size());
        return rank1_pair_impl(_root, left, right);
    }

    void insert(int position, bool value) {
        insert_with_rank(position, value);
    }

    int insert_with_rank(int position, bool value) {
        assert(0 <= position && position <= size());
        int ones_before = 0;
        _root = insert_impl(_root, position, value, ones_before);
        return ones_before;
    }

    bool erase(int position) {
        return erase_with_rank(position).value;
    }

    EraseRankResult erase_with_rank(int position) {
        assert(0 <= position && position < size());
        EraseRankResult result{false, 0};
        _root = erase_impl(_root, position, result);
        return result;
    }
};

}  // namespace dynamic_wavelet_matrix_detail

// A dynamic wavelet matrix for integral sequences.
template <std::integral T>
requires(!std::same_as<std::remove_cv_t<T>, bool>)
class DynamicWaveletMatrix {
   public:
    using value_type = T;
    using unsigned_type = std::make_unsigned_t<T>;

   private:
    static constexpr int bit_width =
        std::numeric_limits<unsigned_type>::digits;
    static constexpr unsigned_type sign_mask = [] {
        if constexpr (std::signed_integral<T>) {
            return unsigned_type(1) << (bit_width - 1);
        } else {
            return unsigned_type(0);
        }
    }();

    int _size = 0;
    std::vector<dynamic_wavelet_matrix_detail::DynamicRankBitVector> _matrix;
    std::array<int, bit_width> _zero_count{};

    static unsigned_type encode(T value) {
        unsigned_type bits;
        if constexpr (std::signed_integral<T>) {
            bits = std::bit_cast<unsigned_type>(value);
        } else {
            bits = value;
        }
        return bits ^ sign_mask;
    }

    static T decode(unsigned_type key) {
        unsigned_type bits = key ^ sign_mask;
        if constexpr (std::signed_integral<T>) {
            return std::bit_cast<T>(bits);
        } else {
            return bits;
        }
    }

    static bool bit(unsigned_type key, int level) {
        return (key >> (bit_width - 1 - level)) & unsigned_type(1);
    }

    void insert_encoded(int position, unsigned_type key) {
        for (int level = 0; level < bit_width; level++) {
            if (bit(key, level)) {
                int ones_before =
                    _matrix[level].insert_with_rank(position, true);
                int next_position = _zero_count[level] + ones_before;
                position = next_position;
            } else {
                int ones_before =
                    _matrix[level].insert_with_rank(position, false);
                int next_position = position - ones_before;
                _zero_count[level]++;
                position = next_position;
            }
        }
        _size++;
    }

    void erase_encoded(int position) {
        for (int level = 0; level < bit_width; level++) {
            auto erased = _matrix[level].erase_with_rank(position);
            int next_position;
            if (erased.value) {
                next_position = _zero_count[level] + erased.ones_before;
            } else {
                next_position = position - erased.ones_before;
                _zero_count[level]--;
            }
            position = next_position;
        }
        _size--;
    }

    int count_less_encoded(int left, int right, unsigned_type upper) const {
        int result = 0;
        for (int level = 0; level < bit_width; level++) {
            auto ranks = _matrix[level].rank1_pair(left, right);
            int left_ones = ranks.left_ones;
            int right_ones = ranks.right_ones;
            if (bit(upper, level)) {
                result += (right - left) - (right_ones - left_ones);
                left = _zero_count[level] + left_ones;
                right = _zero_count[level] + right_ones;
            } else {
                left -= left_ones;
                right -= right_ones;
            }
        }
        return result;
    }

   public:
    DynamicWaveletMatrix() : _matrix(bit_width) {}

    explicit DynamicWaveletMatrix(const std::vector<T>& values)
        : _size(int(values.size())) {
        std::vector<unsigned_type> current(_size);
        std::vector<unsigned_type> next(_size);
        for (int i = 0; i < _size; i++) current[i] = encode(values[i]);

        _matrix.reserve(bit_width);
        for (int level = 0; level < bit_width; level++) {
            std::vector<std::uint8_t> bits(_size);
            int zeros = 0;
            for (int i = 0; i < _size; i++) {
                bits[i] = bit(current[i], level);
                zeros += !bits[i];
            }
            _zero_count[level] = zeros;
            int zero_position = 0;
            int one_position = zeros;
            for (int i = 0; i < _size; i++) {
                if (bits[i]) {
                    next[one_position++] = current[i];
                } else {
                    next[zero_position++] = current[i];
                }
            }
            _matrix.emplace_back(
                bits,
                std::uint32_t(0x9e3779b9U + level * 0x85ebca6bU)
            );
            current.swap(next);
        }
    }

    int size() const {
        return _size;
    }

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

    void clear() {
        *this = DynamicWaveletMatrix();
    }

    T access(int position) const {
        assert(0 <= position && position < _size);
        unsigned_type key = 0;
        for (int level = 0; level < bit_width; level++) {
            auto accessed = _matrix[level].access_with_rank(position);
            if (accessed.value) {
                key |= unsigned_type(1) << (bit_width - 1 - level);
                position = _zero_count[level] + accessed.ones_before;
            } else {
                position -= accessed.ones_before;
            }
        }
        return decode(key);
    }

    T operator[](int position) const {
        return access(position);
    }

    void insert(int position, T value) {
        assert(0 <= position && position <= _size);
        insert_encoded(position, encode(value));
    }

    void push_back(T value) {
        insert(_size, value);
    }

    T erase(int position) {
        assert(0 <= position && position < _size);
        T value = access(position);
        erase_encoded(position);
        return value;
    }

    void set(int position, T value) {
        assert(0 <= position && position < _size);
        if (access(position) == value) return;
        erase_encoded(position);
        insert_encoded(position, encode(value));
    }

    int rank(T value, int right) const {
        assert(0 <= right && right <= _size);
        return rank(value, 0, right);
    }

    int rank(T value, int left, int right) const {
        assert(0 <= left && left <= right && right <= _size);
        unsigned_type key = encode(value);
        for (int level = 0; level < bit_width; level++) {
            auto ranks = _matrix[level].rank1_pair(left, right);
            int left_ones = ranks.left_ones;
            int right_ones = ranks.right_ones;
            if (bit(key, level)) {
                left = _zero_count[level] + left_ones;
                right = _zero_count[level] + right_ones;
            } else {
                left -= left_ones;
                right -= right_ones;
            }
        }
        return right - left;
    }

    T kth_smallest(int left, int right, int k) const {
        assert(0 <= left && left <= right && right <= _size);
        assert(0 <= k && k < right - left);
        unsigned_type key = 0;
        for (int level = 0; level < bit_width; level++) {
            auto ranks = _matrix[level].rank1_pair(left, right);
            int left_ones = ranks.left_ones;
            int right_ones = ranks.right_ones;
            int left_zeros = left - left_ones;
            int right_zeros = right - right_ones;
            int zeros = right_zeros - left_zeros;
            if (k < zeros) {
                left = left_zeros;
                right = right_zeros;
            } else {
                k -= zeros;
                key |= unsigned_type(1) << (bit_width - 1 - level);
                left = _zero_count[level] + left_ones;
                right = _zero_count[level] + right_ones;
            }
        }
        return decode(key);
    }

    T kth_largest(int left, int right, int k) const {
        assert(0 <= left && left <= right && right <= _size);
        assert(0 <= k && k < right - left);
        return kth_smallest(left, right, right - left - 1 - k);
    }

    int range_freq(int left, int right, T upper) const {
        assert(0 <= left && left <= right && right <= _size);
        return count_less_encoded(left, right, encode(upper));
    }

    int range_freq(int left, int right, T lower, T upper) const {
        assert(0 <= left && left <= right && right <= _size);
        if (upper <= lower) return 0;
        return range_freq(left, right, upper) -
               range_freq(left, right, lower);
    }

    std::optional<T> prev_value(int left, int right, T upper) const {
        assert(0 <= left && left <= right && right <= _size);
        int count = range_freq(left, right, upper);
        if (count == 0) return std::nullopt;
        return kth_smallest(left, right, count - 1);
    }

    std::optional<T> next_value(int left, int right, T lower) const {
        assert(0 <= left && left <= right && right <= _size);
        int count = range_freq(left, right, lower);
        if (count == right - left) return std::nullopt;
        return kth_smallest(left, right, count);
    }
};

}  // namespace ds
}  // namespace m1une


#line 17 "ds/wavelet_matrix/dynamic_wavelet_matrix_sum.hpp"

namespace m1une {
namespace ds {

namespace dynamic_wavelet_matrix_sum_detail {

#ifdef M1UNE_DYNAMIC_WAVELET_MATRIX_SUM_PROFILE
struct DynamicWaveletMatrixSumProfile {
    std::uint64_t bitvector_tree_traversals = 0;
    std::uint64_t weight_tree_traversals = 0;
    std::uint64_t chunk_splits = 0;
    std::uint64_t chunk_merges = 0;
    std::uint64_t local_element_moves = 0;
    std::uint64_t full_chunk_rebuilds = 0;
};

inline DynamicWaveletMatrixSumProfile profile;

inline void reset_profile() {
    profile = DynamicWaveletMatrixSumProfile();
}

inline DynamicWaveletMatrixSumProfile get_profile() {
    return profile;
}

#define M1UNE_DWM_SUM_PROFILE_ADD(field, amount) \
    (::m1une::ds::dynamic_wavelet_matrix_sum_detail::profile.field += \
     (amount))
#else
#define M1UNE_DWM_SUM_PROFILE_ADD(field, amount) ((void)0)
#endif

#ifdef M1UNE_DYNAMIC_WAVELET_MATRIX_SUM_CHUNK_CAPACITY
inline constexpr int configured_chunk_capacity =
    M1UNE_DYNAMIC_WAVELET_MATRIX_SUM_CHUNK_CAPACITY;
#else
inline constexpr int configured_chunk_capacity = 64;
#endif

#ifdef M1UNE_DYNAMIC_WAVELET_MATRIX_SUM_GROUP_SIZE
inline constexpr int configured_group_size =
    M1UNE_DYNAMIC_WAVELET_MATRIX_SUM_GROUP_SIZE;
#else
inline constexpr int configured_group_size = 16;
#endif

// A chunked implicit treap storing routing bits and additive weights together.
template <
    typename Sum,
    int ChunkCapacity = configured_chunk_capacity,
    int GroupSize = configured_group_size
>
class DynamicWeightedRankSequence {
   public:
    struct AccessRankResult {
        bool bit;
        Sum weight;
        int ones_before;
    };

    struct EraseRankResult {
        bool bit;
        Sum weight;
        int ones_before;
    };

    struct PrefixStats {
        int ones = 0;
        Sum total_sum{};
        Sum zero_sum{};
    };

    struct PrefixStatsPair {
        PrefixStats left;
        PrefixStats right;
    };

   private:
    static_assert(2 <= ChunkCapacity);
    static_assert(
        ChunkCapacity <= std::numeric_limits<std::uint16_t>::max()
    );
    static_assert(0 < GroupSize && GroupSize <= ChunkCapacity);
    static_assert(ChunkCapacity % GroupSize == 0);
    static constexpr int word_bits = 64;
    static constexpr int word_count =
        (ChunkCapacity + word_bits - 1) / word_bits;
    static constexpr int group_count = ChunkCapacity / GroupSize;
    static constexpr int minimum_chunk_size = ChunkCapacity / 2;

    struct Node {
        std::array<Sum, ChunkCapacity> weights{};
        std::array<Sum, group_count> group_total_sums{};
        std::array<Sum, group_count> group_zero_sums{};
        std::array<std::uint64_t, word_count> bits{};
        Sum chunk_total_sum{};
        Sum chunk_zero_sum{};
        Sum subtree_total_sum{};
        Sum subtree_zero_sum{};
        std::uint32_t priority = 0;
        int left = 0;
        int right = 0;
        int subtree_size = 0;
        int subtree_ones = 0;
        std::uint16_t length = 0;
        std::uint16_t chunk_ones = 0;
    };

    std::vector<Node> _nodes;
    std::vector<int> _free_nodes;
    int _root = 0;
    std::uint32_t _random_state = 1;

    int size_of(int node) const {
        return _nodes[node].subtree_size;
    }

    int ones_of(int node) const {
        return _nodes[node].subtree_ones;
    }

    const Sum& total_sum_of(int node) const {
        return _nodes[node].subtree_total_sum;
    }

    const Sum& zero_sum_of(int node) const {
        return _nodes[node].subtree_zero_sum;
    }

    std::uint32_t next_priority() {
        _random_state ^= _random_state << 13;
        _random_state ^= _random_state >> 17;
        _random_state ^= _random_state << 5;
        if (_random_state == 0) _random_state = 1;
        return _random_state;
    }

    bool local_bit(int node, int position) const {
        return (_nodes[node].bits[position / word_bits] >>
                (position % word_bits)) &
               1U;
    }

    void local_set_bit(int node, int position, bool bit) {
        std::uint64_t mask =
            std::uint64_t(1) << (position % word_bits);
        std::uint64_t& word = _nodes[node].bits[position / word_bits];
        if (bit) {
            word |= mask;
        } else {
            word &= ~mask;
        }
    }

    int local_rank1(int node, int right) const {
        int full_words = right / word_bits;
        int result = 0;
        for (int word = 0; word < full_words; word++) {
            result += std::popcount(_nodes[node].bits[word]);
        }
        int remainder = right % word_bits;
        if (remainder != 0) {
            result += std::popcount(
                _nodes[node].bits[full_words] &
                ((std::uint64_t(1) << remainder) - 1)
            );
        }
        return result;
    }

    void update(int node) {
        if (node == 0) return;
        Node& current = _nodes[node];
        current.subtree_size =
            size_of(current.left) + int(current.length) +
            size_of(current.right);
        current.subtree_ones =
            ones_of(current.left) + int(current.chunk_ones) +
            ones_of(current.right);
        current.subtree_total_sum =
            total_sum_of(current.left) + current.chunk_total_sum +
            total_sum_of(current.right);
        current.subtree_zero_sum =
            zero_sum_of(current.left) + current.chunk_zero_sum +
            zero_sum_of(current.right);
    }

    void rebuild_chunk(int node) {
        M1UNE_DWM_SUM_PROFILE_ADD(full_chunk_rebuilds, 1);
        Node& current = _nodes[node];
        current.group_total_sums.fill(Sum{});
        current.group_zero_sums.fill(Sum{});
        current.chunk_total_sum = Sum{};
        current.chunk_zero_sum = Sum{};
        current.chunk_ones = 0;
        for (int position = 0; position < current.length; position++) {
            int group = position / GroupSize;
            const Sum& weight = current.weights[position];
            current.group_total_sums[group] =
                current.group_total_sums[group] + weight;
            current.chunk_total_sum = current.chunk_total_sum + weight;
            if (local_bit(node, position)) {
                current.chunk_ones++;
            } else {
                current.group_zero_sums[group] =
                    current.group_zero_sums[group] + weight;
                current.chunk_zero_sum = current.chunk_zero_sum + weight;
            }
        }
        update(node);
    }

    PrefixStats local_prefix_stats(int node, int right) const {
        PrefixStats result;
        result.ones = local_rank1(node, right);
        int full_groups = right / GroupSize;
        for (int group = 0; group < full_groups; group++) {
            result.total_sum =
                result.total_sum + _nodes[node].group_total_sums[group];
            result.zero_sum =
                result.zero_sum + _nodes[node].group_zero_sums[group];
        }
        for (int position = full_groups * GroupSize; position < right;
             position++) {
            const Sum& weight = _nodes[node].weights[position];
            result.total_sum = result.total_sum + weight;
            if (!local_bit(node, position)) {
                result.zero_sum = result.zero_sum + weight;
            }
        }
        return result;
    }

    static void add_stats(PrefixStats& destination, const PrefixStats& value) {
        destination.ones += value.ones;
        destination.total_sum = destination.total_sum + value.total_sum;
        destination.zero_sum = destination.zero_sum + value.zero_sum;
    }

    PrefixStats subtree_stats(int node) const {
        return PrefixStats{
            ones_of(node),
            total_sum_of(node),
            zero_sum_of(node)
        };
    }

    void local_insert(int node, int position, bool bit, const Sum& weight) {
        Node& current = _nodes[node];
        assert(0 <= position && position <= current.length);
        assert(current.length < ChunkCapacity);
        M1UNE_DWM_SUM_PROFILE_ADD(
            local_element_moves,
            int(current.length) - position
        );
        for (int i = current.length; position < i; i--) {
            current.weights[i] = current.weights[i - 1];
            local_set_bit(node, i, local_bit(node, i - 1));
        }
        current.weights[position] = weight;
        local_set_bit(node, position, bit);
        current.length++;
        rebuild_chunk(node);
    }

    EraseRankResult local_erase(int node, int position) {
        Node& current = _nodes[node];
        assert(0 <= position && position < current.length);
        M1UNE_DWM_SUM_PROFILE_ADD(
            local_element_moves,
            int(current.length) - position - 1
        );
        EraseRankResult result{
            local_bit(node, position),
            current.weights[position],
            0
        };
        for (int i = position; i + 1 < current.length; i++) {
            current.weights[i] = current.weights[i + 1];
            local_set_bit(node, i, local_bit(node, i + 1));
        }
        current.length--;
        current.weights[current.length] = Sum{};
        local_set_bit(node, current.length, false);
        rebuild_chunk(node);
        return result;
    }

    Sum local_set_weight(int node, int position, const Sum& weight) {
        Node& current = _nodes[node];
        Sum old_weight = current.weights[position];
        Sum delta = weight - old_weight;
        current.weights[position] = weight;
        int group = position / GroupSize;
        current.group_total_sums[group] =
            current.group_total_sums[group] + delta;
        current.chunk_total_sum = current.chunk_total_sum + delta;
        if (!local_bit(node, position)) {
            current.group_zero_sums[group] =
                current.group_zero_sums[group] + delta;
            current.chunk_zero_sum = current.chunk_zero_sum + delta;
        }
        update(node);
        return old_weight;
    }

    Sum local_add_weight(int node, int position, const Sum& delta) {
        return local_set_weight(
            node,
            position,
            _nodes[node].weights[position] + delta
        );
    }

    int new_node() {
        int node;
        if (_free_nodes.empty()) {
            node = int(_nodes.size());
            _nodes.emplace_back();
        } else {
            node = _free_nodes.back();
            _free_nodes.pop_back();
            _nodes[node] = Node();
        }
        _nodes[node].priority = next_priority();
        return node;
    }

    template <std::size_t Capacity>
    void assign_from_values(
        int node,
        const std::array<std::uint8_t, Capacity>& bits,
        const std::array<Sum, Capacity>& weights,
        int first,
        int last
    ) {
        Node& current = _nodes[node];
        current.bits.fill(0);
        current.weights.fill(Sum{});
        current.length = std::uint16_t(last - first);
        for (int position = first; position < last; position++) {
            int destination = position - first;
            current.weights[destination] = weights[position];
            if (bits[position]) local_set_bit(node, destination, true);
        }
        rebuild_chunk(node);
    }

    void assign_from_values(
        int node,
        const std::vector<std::uint8_t>& bits,
        const std::vector<Sum>& weights,
        int first,
        int last
    ) {
        Node& current = _nodes[node];
        current.bits.fill(0);
        current.weights.fill(Sum{});
        current.length = std::uint16_t(last - first);
        for (int position = first; position < last; position++) {
            int destination = position - first;
            current.weights[destination] = weights[position];
            if (bits[position]) local_set_bit(node, destination, true);
        }
        rebuild_chunk(node);
    }

    int new_node(
        const std::vector<std::uint8_t>& bits,
        const std::vector<Sum>& weights,
        int first,
        int last
    ) {
        int node = new_node();
        assign_from_values(node, bits, weights, first, last);
        return node;
    }

    void recycle_node(int node) {
        assert(node != 0);
        _nodes[node] = Node();
        _free_nodes.push_back(node);
    }

    int merge(int left, int right) {
        if (left == 0 || right == 0) return left != 0 ? left : right;
        if (_nodes[left].priority > _nodes[right].priority) {
            _nodes[left].right = merge(_nodes[left].right, right);
            update(left);
            return left;
        }
        _nodes[right].left = merge(left, _nodes[right].left);
        update(right);
        return right;
    }

    int rotate_right(int tree) {
        int result = _nodes[tree].left;
        _nodes[tree].left = _nodes[result].right;
        _nodes[result].right = tree;
        update(tree);
        update(result);
        return result;
    }

    int rotate_left(int tree) {
        int result = _nodes[tree].right;
        _nodes[tree].right = _nodes[result].left;
        _nodes[result].left = tree;
        update(tree);
        update(result);
        return result;
    }

    int extract_leftmost(int tree, int& extracted) {
        if (_nodes[tree].left == 0) {
            extracted = tree;
            int result = _nodes[tree].right;
            _nodes[tree].right = 0;
            update(tree);
            return result;
        }
        _nodes[tree].left = extract_leftmost(_nodes[tree].left, extracted);
        update(tree);
        return tree;
    }

    int extract_rightmost(int tree, int& extracted) {
        if (_nodes[tree].right == 0) {
            extracted = tree;
            int result = _nodes[tree].left;
            _nodes[tree].left = 0;
            update(tree);
            return result;
        }
        _nodes[tree].right = extract_rightmost(_nodes[tree].right, extracted);
        update(tree);
        return tree;
    }

    int rebalance(int node) {
        if (_nodes[node].length >= minimum_chunk_size) return node;

        if (_nodes[node].right != 0) {
            int neighbor = 0;
            _nodes[node].right =
                extract_leftmost(_nodes[node].right, neighbor);
            int node_length = _nodes[node].length;
            int neighbor_length = _nodes[neighbor].length;
            int total = node_length + neighbor_length;
            std::array<std::uint8_t, ChunkCapacity * 2> bits{};
            std::array<Sum, ChunkCapacity * 2> weights{};
            for (int i = 0; i < node_length; i++) {
                bits[i] = local_bit(node, i);
                weights[i] = _nodes[node].weights[i];
            }
            for (int i = 0; i < neighbor_length; i++) {
                bits[node_length + i] = local_bit(neighbor, i);
                weights[node_length + i] = _nodes[neighbor].weights[i];
            }
            if (total <= ChunkCapacity) {
                M1UNE_DWM_SUM_PROFILE_ADD(chunk_merges, 1);
                assign_from_values(node, bits, weights, 0, total);
                recycle_node(neighbor);
            } else {
                int left_length = total / 2;
                assign_from_values(node, bits, weights, 0, left_length);
                assign_from_values(
                    neighbor,
                    bits,
                    weights,
                    left_length,
                    total
                );
                _nodes[node].right = merge(neighbor, _nodes[node].right);
            }
            update(node);
            return node;
        }

        if (_nodes[node].left != 0) {
            int neighbor = 0;
            _nodes[node].left =
                extract_rightmost(_nodes[node].left, neighbor);
            int neighbor_length = _nodes[neighbor].length;
            int node_length = _nodes[node].length;
            int total = neighbor_length + node_length;
            std::array<std::uint8_t, ChunkCapacity * 2> bits{};
            std::array<Sum, ChunkCapacity * 2> weights{};
            for (int i = 0; i < neighbor_length; i++) {
                bits[i] = local_bit(neighbor, i);
                weights[i] = _nodes[neighbor].weights[i];
            }
            for (int i = 0; i < node_length; i++) {
                bits[neighbor_length + i] = local_bit(node, i);
                weights[neighbor_length + i] = _nodes[node].weights[i];
            }
            if (total <= ChunkCapacity) {
                M1UNE_DWM_SUM_PROFILE_ADD(chunk_merges, 1);
                assign_from_values(node, bits, weights, 0, total);
                recycle_node(neighbor);
            } else {
                int left_length = total / 2;
                assign_from_values(
                    neighbor,
                    bits,
                    weights,
                    0,
                    left_length
                );
                assign_from_values(
                    node,
                    bits,
                    weights,
                    left_length,
                    total
                );
                _nodes[node].left = merge(_nodes[node].left, neighbor);
            }
            update(node);
        }
        return node;
    }

    int insert_impl(
        int tree,
        int position,
        bool bit,
        const Sum& weight,
        int& ones_before
    ) {
        if (tree == 0) {
            int node = new_node();
            local_insert(node, 0, bit, weight);
            return node;
        }

        int left_size = size_of(_nodes[tree].left);
        int length = _nodes[tree].length;
        if (position < left_size) {
            _nodes[tree].left = insert_impl(
                _nodes[tree].left,
                position,
                bit,
                weight,
                ones_before
            );
            update(tree);
            if (_nodes[_nodes[tree].left].priority >
                _nodes[tree].priority) {
                tree = rotate_right(tree);
            }
            return tree;
        }
        if (position > left_size + length) {
            ones_before +=
                ones_of(_nodes[tree].left) + _nodes[tree].chunk_ones;
            _nodes[tree].right = insert_impl(
                _nodes[tree].right,
                position - left_size - length,
                bit,
                weight,
                ones_before
            );
            update(tree);
            if (_nodes[_nodes[tree].right].priority >
                _nodes[tree].priority) {
                tree = rotate_left(tree);
            }
            return tree;
        }

        int local_position = position - left_size;
        ones_before += ones_of(_nodes[tree].left) +
                       local_rank1(tree, local_position);
        if (length < ChunkCapacity) {
            local_insert(tree, local_position, bit, weight);
            return tree;
        }

        std::array<std::uint8_t, ChunkCapacity> bits{};
        std::array<Sum, ChunkCapacity> weights{};
        M1UNE_DWM_SUM_PROFILE_ADD(chunk_splits, 1);
        for (int i = 0; i < ChunkCapacity; i++) {
            bits[i] = local_bit(tree, i);
            weights[i] = _nodes[tree].weights[i];
        }
        int right_chunk = new_node();
        int middle = ChunkCapacity / 2;
        assign_from_values(tree, bits, weights, 0, middle);
        assign_from_values(
            right_chunk,
            bits,
            weights,
            middle,
            ChunkCapacity
        );
        if (local_position <= middle) {
            local_insert(tree, local_position, bit, weight);
        } else {
            local_insert(
                right_chunk,
                local_position - middle,
                bit,
                weight
            );
        }

        int old_right = _nodes[tree].right;
        _nodes[tree].right = 0;
        update(tree);
        return merge(merge(tree, right_chunk), old_right);
    }

    int erase_impl(
        int tree,
        int position,
        EraseRankResult& result
    ) {
        int left_size = size_of(_nodes[tree].left);
        int length = _nodes[tree].length;
        if (position < left_size) {
            _nodes[tree].left = erase_impl(
                _nodes[tree].left,
                position,
                result
            );
            update(tree);
            return tree;
        }
        if (position >= left_size + length) {
            result.ones_before +=
                ones_of(_nodes[tree].left) + _nodes[tree].chunk_ones;
            _nodes[tree].right = erase_impl(
                _nodes[tree].right,
                position - left_size - length,
                result
            );
            update(tree);
            return tree;
        }

        int local_position = position - left_size;
        result.ones_before += ones_of(_nodes[tree].left) +
                              local_rank1(tree, local_position);
        EraseRankResult local = local_erase(tree, local_position);
        result.bit = local.bit;
        result.weight = local.weight;
        if (_nodes[tree].length == 0) {
            int merged = merge(_nodes[tree].left, _nodes[tree].right);
            recycle_node(tree);
            return merged;
        }
        return rebalance(tree);
    }

    template <bool Add>
    void change_weight_impl(
        int tree,
        int position,
        const Sum& value,
        AccessRankResult& result
    ) {
        int left_size = size_of(_nodes[tree].left);
        if (position < left_size) {
            change_weight_impl<Add>(
                _nodes[tree].left,
                position,
                value,
                result
            );
        } else if (position < left_size + _nodes[tree].length) {
            int local_position = position - left_size;
            result.ones_before += ones_of(_nodes[tree].left) +
                                  local_rank1(tree, local_position);
            result.bit = local_bit(tree, local_position);
            if constexpr (Add) {
                result.weight = local_add_weight(tree, local_position, value);
            } else {
                result.weight = local_set_weight(tree, local_position, value);
            }
            return;
        } else {
            result.ones_before +=
                ones_of(_nodes[tree].left) + _nodes[tree].chunk_ones;
            change_weight_impl<Add>(
                _nodes[tree].right,
                position - left_size - _nodes[tree].length,
                value,
                result
            );
        }
        update(tree);
    }

    PrefixStats prefix_stats_impl(int tree, int right) const {
        PrefixStats result;
        while (tree != 0 && right != 0) {
            int left_size = size_of(_nodes[tree].left);
            if (right <= left_size) {
                tree = _nodes[tree].left;
                continue;
            }
            add_stats(result, subtree_stats(_nodes[tree].left));
            right -= left_size;
            int take = std::min(right, int(_nodes[tree].length));
            add_stats(result, local_prefix_stats(tree, take));
            right -= take;
            if (right == 0) break;
            tree = _nodes[tree].right;
        }
        return result;
    }

    PrefixStatsPair prefix_stats_pair_impl(
        int tree,
        int left,
        int right
    ) const {
        if (left == right) {
            PrefixStats value = prefix_stats_impl(tree, left);
            return PrefixStatsPair{value, value};
        }
        if (tree == 0 || right == 0) return PrefixStatsPair{};

        int left_size = size_of(_nodes[tree].left);
        int chunk_end = left_size + _nodes[tree].length;
        if (right <= left_size) {
            return prefix_stats_pair_impl(
                _nodes[tree].left,
                left,
                right
            );
        }
        if (chunk_end <= left) {
            PrefixStats base = subtree_stats(_nodes[tree].left);
            add_stats(base, local_prefix_stats(tree, _nodes[tree].length));
            PrefixStatsPair result = prefix_stats_pair_impl(
                _nodes[tree].right,
                left - chunk_end,
                right - chunk_end
            );
            add_stats(result.left, base);
            add_stats(result.right, base);
            return result;
        }

        PrefixStats left_stats;
        if (left <= left_size) {
            left_stats = prefix_stats_impl(_nodes[tree].left, left);
        } else {
            left_stats = subtree_stats(_nodes[tree].left);
            add_stats(
                left_stats,
                local_prefix_stats(tree, left - left_size)
            );
        }

        PrefixStats right_stats = subtree_stats(_nodes[tree].left);
        if (right <= chunk_end) {
            add_stats(
                right_stats,
                local_prefix_stats(tree, right - left_size)
            );
        } else {
            add_stats(
                right_stats,
                local_prefix_stats(tree, _nodes[tree].length)
            );
            add_stats(
                right_stats,
                prefix_stats_impl(_nodes[tree].right, right - chunk_end)
            );
        }
        return PrefixStatsPair{left_stats, right_stats};
    }

    template <class Predicate>
    int consume_chunk_prefix(
        int node,
        int left,
        int right,
        Sum& sum,
        Predicate& predicate
    ) const {
        int position = left;
        while (position < right) {
            if (position % GroupSize == 0 &&
                position + GroupSize <= right) {
                int group = position / GroupSize;
                Sum candidate =
                    sum + _nodes[node].group_total_sums[group];
                if (predicate(candidate)) {
                    sum = candidate;
                    position += GroupSize;
                    continue;
                }
            }
            Sum candidate = sum + _nodes[node].weights[position];
            if (!predicate(candidate)) break;
            sum = candidate;
            position++;
        }
        return position - left;
    }

    template <class Predicate>
    int consume_chunk_suffix(
        int node,
        int left,
        int right,
        Sum& sum,
        Predicate& predicate
    ) const {
        int position = right;
        while (left < position) {
            if (position % GroupSize == 0 &&
                left <= position - GroupSize) {
                int group = position / GroupSize - 1;
                Sum candidate =
                    sum + _nodes[node].group_total_sums[group];
                if (predicate(candidate)) {
                    sum = candidate;
                    position -= GroupSize;
                    continue;
                }
            }
            Sum candidate = sum + _nodes[node].weights[position - 1];
            if (!predicate(candidate)) break;
            sum = candidate;
            position--;
        }
        return right - position;
    }

    template <class Predicate>
    int max_prefix_impl(
        int tree,
        int left,
        int right,
        Sum& sum,
        Predicate& predicate
    ) const {
        assert(tree != 0);
        assert(0 <= left && left < right && right <= size_of(tree));
        if (left == 0 && right == size_of(tree)) {
            Sum candidate = sum + total_sum_of(tree);
            if (predicate(candidate)) {
                sum = candidate;
                return size_of(tree);
            }
        }

        int left_size = size_of(_nodes[tree].left);
        int chunk_end = left_size + _nodes[tree].length;
        int count = 0;
        if (left < left_size) {
            int subtree_right = std::min(right, left_size);
            int consumed = max_prefix_impl(
                _nodes[tree].left,
                left,
                subtree_right,
                sum,
                predicate
            );
            count += consumed;
            if (consumed < subtree_right - left) return count;
        }

        int chunk_left = std::max(left, left_size);
        int chunk_right = std::min(right, chunk_end);
        if (chunk_left < chunk_right) {
            int consumed = consume_chunk_prefix(
                tree,
                chunk_left - left_size,
                chunk_right - left_size,
                sum,
                predicate
            );
            count += consumed;
            if (consumed < chunk_right - chunk_left) return count;
        }

        if (chunk_end < right) {
            int subtree_left = std::max(left, chunk_end) - chunk_end;
            int subtree_right = right - chunk_end;
            count += max_prefix_impl(
                _nodes[tree].right,
                subtree_left,
                subtree_right,
                sum,
                predicate
            );
        }
        return count;
    }

    template <class Predicate>
    int max_suffix_impl(
        int tree,
        int left,
        int right,
        Sum& sum,
        Predicate& predicate
    ) const {
        assert(tree != 0);
        assert(0 <= left && left < right && right <= size_of(tree));
        if (left == 0 && right == size_of(tree)) {
            Sum candidate = sum + total_sum_of(tree);
            if (predicate(candidate)) {
                sum = candidate;
                return size_of(tree);
            }
        }

        int left_size = size_of(_nodes[tree].left);
        int chunk_end = left_size + _nodes[tree].length;
        int count = 0;
        if (chunk_end < right) {
            int subtree_left = std::max(left, chunk_end) - chunk_end;
            int subtree_right = right - chunk_end;
            int consumed = max_suffix_impl(
                _nodes[tree].right,
                subtree_left,
                subtree_right,
                sum,
                predicate
            );
            count += consumed;
            if (consumed < subtree_right - subtree_left) return count;
        }

        int chunk_left = std::max(left, left_size);
        int chunk_right = std::min(right, chunk_end);
        if (chunk_left < chunk_right) {
            int consumed = consume_chunk_suffix(
                tree,
                chunk_left - left_size,
                chunk_right - left_size,
                sum,
                predicate
            );
            count += consumed;
            if (consumed < chunk_right - chunk_left) return count;
        }

        if (left < left_size) {
            int subtree_right = std::min(right, left_size);
            count += max_suffix_impl(
                _nodes[tree].left,
                left,
                subtree_right,
                sum,
                predicate
            );
        }
        return count;
    }

    void update_subtree(int tree) {
        if (tree == 0) return;
        update_subtree(_nodes[tree].left);
        update_subtree(_nodes[tree].right);
        update(tree);
    }

    void build(
        const std::vector<std::uint8_t>& bits,
        const std::vector<Sum>& weights
    ) {
        assert(bits.size() == weights.size());
        _nodes.clear();
        _nodes.emplace_back();
        _free_nodes.clear();
        _root = 0;
        _nodes.reserve(weights.size() / minimum_chunk_size + 2);

        std::vector<int> stack;
        for (int first = 0; first < int(weights.size());
             first += ChunkCapacity) {
            int last =
                std::min(first + ChunkCapacity, int(weights.size()));
            int node = new_node(bits, weights, first, last);
            int left = 0;
            while (!stack.empty() &&
                   _nodes[stack.back()].priority < _nodes[node].priority) {
                left = stack.back();
                stack.pop_back();
            }
            _nodes[node].left = left;
            if (!stack.empty()) _nodes[stack.back()].right = node;
            stack.push_back(node);
        }
        if (!stack.empty()) _root = stack.front();
        update_subtree(_root);
    }

   public:
    DynamicWeightedRankSequence() : _nodes(1) {}

    DynamicWeightedRankSequence(
        const std::vector<std::uint8_t>& bits,
        const std::vector<Sum>& weights,
        std::uint32_t seed = 1
    ) : _random_state(seed == 0 ? 1 : seed) {
        build(bits, weights);
    }

    explicit DynamicWeightedRankSequence(
        const std::vector<Sum>& weights,
        std::uint32_t seed = 1
    ) : _random_state(seed == 0 ? 1 : seed) {
        build(std::vector<std::uint8_t>(weights.size()), weights);
    }

    int size() const {
        return size_of(_root);
    }

    AccessRankResult access_with_rank(int position) const {
        assert(0 <= position && position < size());
        M1UNE_DWM_SUM_PROFILE_ADD(weight_tree_traversals, 1);
        int tree = _root;
        int ones_before = 0;
        while (tree != 0) {
            int left_size = size_of(_nodes[tree].left);
            if (position < left_size) {
                tree = _nodes[tree].left;
            } else if (position < left_size + _nodes[tree].length) {
                int local_position = position - left_size;
                ones_before += ones_of(_nodes[tree].left) +
                               local_rank1(tree, local_position);
                return AccessRankResult{
                    local_bit(tree, local_position),
                    _nodes[tree].weights[local_position],
                    ones_before
                };
            } else {
                ones_before +=
                    ones_of(_nodes[tree].left) + _nodes[tree].chunk_ones;
                position -= left_size + _nodes[tree].length;
                tree = _nodes[tree].right;
            }
        }
        assert(false);
        return AccessRankResult{false, Sum{}, 0};
    }

    PrefixStats prefix_stats(int right) const {
        assert(0 <= right && right <= size());
        M1UNE_DWM_SUM_PROFILE_ADD(weight_tree_traversals, 1);
        return prefix_stats_impl(_root, right);
    }

    PrefixStatsPair prefix_stats_pair(int left, int right) const {
        assert(0 <= left && left <= right && right <= size());
        M1UNE_DWM_SUM_PROFILE_ADD(weight_tree_traversals, 1);
        return prefix_stats_pair_impl(_root, left, right);
    }

    Sum range_sum(int left, int right) const {
        PrefixStatsPair stats = prefix_stats_pair(left, right);
        return stats.right.total_sum - stats.left.total_sum;
    }

    int insert_with_rank(
        int position,
        bool bit,
        const Sum& weight
    ) {
        assert(0 <= position && position <= size());
        M1UNE_DWM_SUM_PROFILE_ADD(weight_tree_traversals, 1);
        int ones_before = 0;
        _root = insert_impl(
            _root,
            position,
            bit,
            weight,
            ones_before
        );
        return ones_before;
    }

    EraseRankResult erase_with_rank(int position) {
        assert(0 <= position && position < size());
        M1UNE_DWM_SUM_PROFILE_ADD(weight_tree_traversals, 1);
        EraseRankResult result{false, Sum{}, 0};
        _root = erase_impl(_root, position, result);
        return result;
    }

    AccessRankResult set_weight_with_rank(
        int position,
        const Sum& weight
    ) {
        assert(0 <= position && position < size());
        M1UNE_DWM_SUM_PROFILE_ADD(weight_tree_traversals, 1);
        AccessRankResult result{false, Sum{}, 0};
        change_weight_impl<false>(_root, position, weight, result);
        return result;
    }

    AccessRankResult add_weight_with_rank(
        int position,
        const Sum& delta
    ) {
        assert(0 <= position && position < size());
        M1UNE_DWM_SUM_PROFILE_ADD(weight_tree_traversals, 1);
        AccessRankResult result{false, Sum{}, 0};
        change_weight_impl<true>(_root, position, delta, result);
        return result;
    }

    template <class Predicate>
    int max_prefix(
        int left,
        int right,
        Sum& sum,
        Predicate& predicate
    ) const {
        assert(0 <= left && left <= right && right <= size());
        if (left == right) return 0;
        M1UNE_DWM_SUM_PROFILE_ADD(weight_tree_traversals, 1);
        return max_prefix_impl(_root, left, right, sum, predicate);
    }

    template <class Predicate>
    int max_suffix(
        int left,
        int right,
        Sum& sum,
        Predicate& predicate
    ) const {
        assert(0 <= left && left <= right && right <= size());
        if (left == right) return 0;
        M1UNE_DWM_SUM_PROFILE_ADD(weight_tree_traversals, 1);
        return max_suffix_impl(_root, left, right, sum, predicate);
    }
};

}  // namespace dynamic_wavelet_matrix_sum_detail

#undef M1UNE_DWM_SUM_PROFILE_ADD

// A dynamic wavelet matrix with additive weights.
// By default, each value is also used as its weight.
template <
    std::integral T,
    typename Sum = T,
    int BitWidth = std::numeric_limits<std::make_unsigned_t<T>>::digits
>
requires(!std::same_as<std::remove_cv_t<T>, bool>)
class DynamicWaveletMatrixSum {
   public:
    using value_type = T;
    using sum_type = Sum;
    using unsigned_type = std::make_unsigned_t<T>;

   private:
    static constexpr int full_bit_width =
        std::numeric_limits<unsigned_type>::digits;
    static_assert(1 <= BitWidth && BitWidth <= full_bit_width);
    static_assert(
        BitWidth == full_bit_width || std::unsigned_integral<T>,
        "reduced-width keys must use an unsigned type"
    );

    static constexpr unsigned_type sign_mask = [] {
        if constexpr (std::signed_integral<T>) {
            return unsigned_type(1) << (BitWidth - 1);
        } else {
            return unsigned_type(0);
        }
    }();

    static constexpr unsigned_type reduced_limit = [] {
        if constexpr (BitWidth < full_bit_width) {
            return unsigned_type(1) << BitWidth;
        } else {
            return unsigned_type(0);
        }
    }();

    using Level = dynamic_wavelet_matrix_sum_detail::
        DynamicWeightedRankSequence<Sum>;

    struct ErasedElement {
        unsigned_type key;
        Sum weight;
    };

    int _size = 0;
    std::vector<Level> _levels;
    Level _final_weights;
    std::array<int, BitWidth> _zero_count{};

    static bool key_fits(T value) {
        if constexpr (BitWidth < full_bit_width) {
            return static_cast<unsigned_type>(value) < reduced_limit;
        } else {
            return true;
        }
    }

    static unsigned_type encode_unchecked(T value) {
        unsigned_type bits;
        if constexpr (std::signed_integral<T>) {
            bits = std::bit_cast<unsigned_type>(value);
        } else {
            bits = value;
        }
        return bits ^ sign_mask;
    }

    static unsigned_type encode_key(T value) {
        assert(key_fits(value));
        return encode_unchecked(value);
    }

    static T decode(unsigned_type key) {
        unsigned_type bits = key ^ sign_mask;
        if constexpr (std::signed_integral<T>) {
            return std::bit_cast<T>(bits);
        } else {
            return bits;
        }
    }

    static bool bit(unsigned_type key, int level) {
        return (key >> (BitWidth - 1 - level)) & unsigned_type(1);
    }

    static Sum range_total(const typename Level::PrefixStatsPair& stats) {
        return stats.right.total_sum - stats.left.total_sum;
    }

    static Sum range_zero(const typename Level::PrefixStatsPair& stats) {
        return stats.right.zero_sum - stats.left.zero_sum;
    }

    void build(const std::vector<T>& values, const std::vector<Sum>& weights) {
        assert(values.size() == weights.size());
        _size = int(values.size());

        std::vector<unsigned_type> current_keys(_size);
        std::vector<unsigned_type> next_keys(_size);
        std::vector<Sum> current_weights(weights);
        std::vector<Sum> next_weights(_size);
        for (int i = 0; i < _size; i++) {
            current_keys[i] = encode_key(values[i]);
        }

        _levels.clear();
        _levels.reserve(BitWidth);
        for (int level = 0; level < BitWidth; level++) {
            std::vector<std::uint8_t> bits(_size);
            int zeros = 0;
            for (int i = 0; i < _size; i++) {
                bits[i] = bit(current_keys[i], level);
                zeros += !bits[i];
            }
            _zero_count[level] = zeros;
            _levels.emplace_back(
                bits,
                current_weights,
                std::uint32_t(0x9e3779b9U + level * 0x85ebca6bU)
            );

            int zero_position = 0;
            int one_position = zeros;
            for (int i = 0; i < _size; i++) {
                int next_position = bits[i] ? one_position++ : zero_position++;
                next_keys[next_position] = current_keys[i];
                next_weights[next_position] = current_weights[i];
            }
            current_keys.swap(next_keys);
            current_weights.swap(next_weights);
        }
        _final_weights = Level(current_weights, 0xb7e15162U);
    }

    void insert_encoded(int position, unsigned_type key, const Sum& weight) {
        for (int level = 0; level < BitWidth; level++) {
            bool one = bit(key, level);
            int ones_before =
                _levels[level].insert_with_rank(position, one, weight);
            int next_position;
            if (one) {
                next_position = _zero_count[level] + ones_before;
            } else {
                next_position = position - ones_before;
                _zero_count[level]++;
            }
            position = next_position;
        }
        _final_weights.insert_with_rank(position, false, weight);
        _size++;
    }

    ErasedElement erase_encoded(int position) {
        unsigned_type key = 0;
        Sum weight{};
        for (int level = 0; level < BitWidth; level++) {
            auto erased = _levels[level].erase_with_rank(position);
            if (level == 0) weight = erased.weight;
            int next_position;
            if (erased.bit) {
                key |= unsigned_type(1) << (BitWidth - 1 - level);
                next_position = _zero_count[level] + erased.ones_before;
            } else {
                next_position = position - erased.ones_before;
                _zero_count[level]--;
            }
            position = next_position;
        }
        _final_weights.erase_with_rank(position);
        _size--;
        return ErasedElement{key, weight};
    }

    int count_less_encoded(int left, int right, unsigned_type upper) const {
        int result = 0;
        for (int level = 0; level < BitWidth; level++) {
            auto stats = _levels[level].prefix_stats_pair(left, right);
            int left_ones = stats.left.ones;
            int right_ones = stats.right.ones;
            if (bit(upper, level)) {
                result += (right - left) - (right_ones - left_ones);
                left = _zero_count[level] + left_ones;
                right = _zero_count[level] + right_ones;
            } else {
                left -= left_ones;
                right -= right_ones;
            }
        }
        return result;
    }

    int count_less(int left, int right, T upper) const {
        if constexpr (BitWidth < full_bit_width) {
            if (!key_fits(upper)) return right - left;
        }
        return count_less_encoded(left, right, encode_unchecked(upper));
    }

    Sum sum_less_encoded(int left, int right, unsigned_type upper) const {
        Sum result{};
        for (int level = 0; level < BitWidth; level++) {
            auto stats = _levels[level].prefix_stats_pair(left, right);
            int left_ones = stats.left.ones;
            int right_ones = stats.right.ones;
            if (bit(upper, level)) {
                result = result + range_zero(stats);
                left = _zero_count[level] + left_ones;
                right = _zero_count[level] + right_ones;
            } else {
                left -= left_ones;
                right -= right_ones;
            }
        }
        return result;
    }

    Sum sum_less(int left, int right, T upper) const {
        if constexpr (BitWidth < full_bit_width) {
            if (!key_fits(upper)) return range_sum(left, right);
        }
        return sum_less_encoded(left, right, encode_unchecked(upper));
    }

   public:
    DynamicWaveletMatrixSum() : _levels(BitWidth) {}

    explicit DynamicWaveletMatrixSum(const std::vector<T>& values)
        requires std::convertible_to<T, Sum>
    {
        std::vector<Sum> weights;
        weights.reserve(values.size());
        for (T value : values) weights.push_back(static_cast<Sum>(value));
        build(values, weights);
    }

    DynamicWaveletMatrixSum(
        const std::vector<T>& values,
        const std::vector<Sum>& weights
    ) {
        build(values, weights);
    }

    int size() const {
        return _size;
    }

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

    void clear() {
        *this = DynamicWaveletMatrixSum();
    }

    T access(int position) const {
        assert(0 <= position && position < _size);
        unsigned_type key = 0;
        for (int level = 0; level < BitWidth; level++) {
            auto accessed = _levels[level].access_with_rank(position);
            if (accessed.bit) {
                key |= unsigned_type(1) << (BitWidth - 1 - level);
                position = _zero_count[level] + accessed.ones_before;
            } else {
                position -= accessed.ones_before;
            }
        }
        return decode(key);
    }

    T operator[](int position) const {
        return access(position);
    }

    Sum weight(int position) const {
        assert(0 <= position && position < _size);
        return _levels[0].access_with_rank(position).weight;
    }

    std::pair<T, Sum> get(int position) const {
        assert(0 <= position && position < _size);
        unsigned_type key = 0;
        Sum result_weight{};
        for (int level = 0; level < BitWidth; level++) {
            auto accessed = _levels[level].access_with_rank(position);
            if (level == 0) result_weight = accessed.weight;
            if (accessed.bit) {
                key |= unsigned_type(1) << (BitWidth - 1 - level);
                position = _zero_count[level] + accessed.ones_before;
            } else {
                position -= accessed.ones_before;
            }
        }
        return std::pair<T, Sum>(decode(key), result_weight);
    }

    void insert(int position, T value)
        requires std::convertible_to<T, Sum>
    {
        insert(position, value, static_cast<Sum>(value));
    }

    void insert(int position, T value, const Sum& weight) {
        assert(0 <= position && position <= _size);
        insert_encoded(position, encode_key(value), weight);
    }

    void push_back(T value)
        requires std::convertible_to<T, Sum>
    {
        insert(_size, value);
    }

    void push_back(T value, const Sum& weight) {
        insert(_size, value, weight);
    }

    std::pair<T, Sum> erase(int position) {
        assert(0 <= position && position < _size);
        ErasedElement erased = erase_encoded(position);
        return std::pair<T, Sum>(decode(erased.key), erased.weight);
    }

    void set(int position, T value)
        requires std::convertible_to<T, Sum>
    {
        set(position, value, static_cast<Sum>(value));
    }

    void set(int position, T value, const Sum& weight) {
        assert(0 <= position && position < _size);
        unsigned_type key = encode_key(value);
        erase_encoded(position);
        insert_encoded(position, key, weight);
    }

    void set_value(int position, T value) {
        assert(0 <= position && position < _size);
        unsigned_type key = encode_key(value);
        ErasedElement erased = erase_encoded(position);
        insert_encoded(position, key, erased.weight);
    }

    void set_weight(int position, const Sum& new_weight) {
        assert(0 <= position && position < _size);
        for (int level = 0; level < BitWidth; level++) {
            auto accessed =
                _levels[level].set_weight_with_rank(position, new_weight);
            if (accessed.bit) {
                position = _zero_count[level] + accessed.ones_before;
            } else {
                position -= accessed.ones_before;
            }
        }
        _final_weights.set_weight_with_rank(position, new_weight);
    }

    void add_weight(int position, const Sum& delta) {
        assert(0 <= position && position < _size);
        for (int level = 0; level < BitWidth; level++) {
            auto accessed =
                _levels[level].add_weight_with_rank(position, delta);
            if (accessed.bit) {
                position = _zero_count[level] + accessed.ones_before;
            } else {
                position -= accessed.ones_before;
            }
        }
        _final_weights.add_weight_with_rank(position, delta);
    }

    int rank(T value, int right) const {
        assert(0 <= right && right <= _size);
        return rank(value, 0, right);
    }

    int rank(T value, int left, int right) const {
        assert(0 <= left && left <= right && right <= _size);
        if constexpr (BitWidth < full_bit_width) {
            if (!key_fits(value)) return 0;
        }
        unsigned_type key = encode_unchecked(value);
        for (int level = 0; level < BitWidth; level++) {
            auto stats = _levels[level].prefix_stats_pair(left, right);
            int left_ones = stats.left.ones;
            int right_ones = stats.right.ones;
            if (bit(key, level)) {
                left = _zero_count[level] + left_ones;
                right = _zero_count[level] + right_ones;
            } else {
                left -= left_ones;
                right -= right_ones;
            }
        }
        return right - left;
    }

    T kth_smallest(int left, int right, int k) const {
        assert(0 <= left && left <= right && right <= _size);
        assert(0 <= k && k < right - left);
        unsigned_type key = 0;
        for (int level = 0; level < BitWidth; level++) {
            auto stats = _levels[level].prefix_stats_pair(left, right);
            int left_ones = stats.left.ones;
            int right_ones = stats.right.ones;
            int left_zeros = left - left_ones;
            int right_zeros = right - right_ones;
            int zeros = right_zeros - left_zeros;
            if (k < zeros) {
                left = left_zeros;
                right = right_zeros;
            } else {
                k -= zeros;
                key |= unsigned_type(1) << (BitWidth - 1 - level);
                left = _zero_count[level] + left_ones;
                right = _zero_count[level] + right_ones;
            }
        }
        return decode(key);
    }

    T kth_largest(int left, int right, int k) const {
        assert(0 <= left && left <= right && right <= _size);
        assert(0 <= k && k < right - left);
        return kth_smallest(left, right, right - left - 1 - k);
    }

    int range_freq(int left, int right, T upper) const {
        assert(0 <= left && left <= right && right <= _size);
        return count_less(left, right, upper);
    }

    int range_freq(int left, int right, T lower, T upper) const {
        assert(0 <= left && left <= right && right <= _size);
        if (upper <= lower) return 0;
        return count_less(left, right, upper) -
               count_less(left, right, lower);
    }

    std::optional<T> prev_value(int left, int right, T upper) const {
        assert(0 <= left && left <= right && right <= _size);
        int count = range_freq(left, right, upper);
        if (count == 0) return std::nullopt;
        return kth_smallest(left, right, count - 1);
    }

    std::optional<T> next_value(int left, int right, T lower) const {
        assert(0 <= left && left <= right && right <= _size);
        int count = range_freq(left, right, lower);
        if (count == right - left) return std::nullopt;
        return kth_smallest(left, right, count);
    }

    Sum range_sum(int left, int right) const {
        assert(0 <= left && left <= right && right <= _size);
        return _levels[0].range_sum(left, right);
    }

    Sum range_sum(int left, int right, T upper) const {
        assert(0 <= left && left <= right && right <= _size);
        return sum_less(left, right, upper);
    }

    Sum range_sum(int left, int right, T lower, T upper) const {
        assert(0 <= left && left <= right && right <= _size);
        if (upper <= lower) return Sum{};
        return sum_less(left, right, upper) -
               sum_less(left, right, lower);
    }

    Sum sum_k_smallest(int left, int right, int k) const {
        assert(0 <= left && left <= right && right <= _size);
        assert(0 <= k && k <= right - left);
        Sum result{};
        for (int level = 0; level < BitWidth; level++) {
            auto stats = _levels[level].prefix_stats_pair(left, right);
            int left_ones = stats.left.ones;
            int right_ones = stats.right.ones;
            int left_zeros = left - left_ones;
            int right_zeros = right - right_ones;
            int zeros = right_zeros - left_zeros;
            if (k < zeros) {
                left = left_zeros;
                right = right_zeros;
            } else {
                result = result + range_zero(stats);
                k -= zeros;
                left = _zero_count[level] + left_ones;
                right = _zero_count[level] + right_ones;
            }
        }
        return result + _final_weights.range_sum(left, left + k);
    }

    Sum sum_k_largest(int left, int right, int k) const {
        assert(0 <= left && left <= right && right <= _size);
        assert(0 <= k && k <= right - left);
        return range_sum(left, right) -
               sum_k_smallest(left, right, right - left - k);
    }

    template <class Predicate>
    int max_count_smallest(
        int left,
        int right,
        Predicate predicate
    ) const {
        assert(0 <= left && left <= right && right <= _size);
        assert(predicate(Sum{}));
        Sum result{};
        int count = 0;
        for (int level = 0; level < BitWidth; level++) {
            auto stats = _levels[level].prefix_stats_pair(left, right);
            int left_ones = stats.left.ones;
            int right_ones = stats.right.ones;
            int left_zeros = left - left_ones;
            int right_zeros = right - right_ones;
            int zeros = right_zeros - left_zeros;
            Sum candidate = result + range_zero(stats);
            if (predicate(candidate)) {
                result = candidate;
                count += zeros;
                left = _zero_count[level] + left_ones;
                right = _zero_count[level] + right_ones;
            } else {
                left = left_zeros;
                right = right_zeros;
            }
        }
        return count +
               _final_weights.max_prefix(
                   left,
                   right,
                   result,
                   predicate
               );
    }

    template <class Predicate>
    int max_count_largest(
        int left,
        int right,
        Predicate predicate
    ) const {
        assert(0 <= left && left <= right && right <= _size);
        assert(predicate(Sum{}));
        Sum result{};
        int count = 0;
        for (int level = 0; level < BitWidth; level++) {
            auto stats = _levels[level].prefix_stats_pair(left, right);
            int left_ones = stats.left.ones;
            int right_ones = stats.right.ones;
            int left_zeros = left - left_ones;
            int right_zeros = right - right_ones;
            int ones = right_ones - left_ones;
            Sum zero_result = range_zero(stats);
            Sum one_result = range_total(stats) - zero_result;
            Sum candidate = result + one_result;
            if (predicate(candidate)) {
                result = candidate;
                count += ones;
                left = left_zeros;
                right = right_zeros;
            } else {
                left = _zero_count[level] + left_ones;
                right = _zero_count[level] + right_ones;
            }
        }
        return count +
               _final_weights.max_suffix(
                   left,
                   right,
                   result,
                   predicate
               );
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "utilities/fast_io.hpp"



#line 6 "utilities/fast_io.hpp"
#include <cerrno>
#include <charconv>
#include <cstddef>
#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 5 "verify/ds/wavelet_matrix/dynamic_wavelet_matrix_sum.test.cpp"

#line 13 "verify/ds/wavelet_matrix/dynamic_wavelet_matrix_sum.test.cpp"

namespace {

using Matrix = m1une::ds::DynamicWaveletMatrixSum<int, long long>;

std::uint64_t random_state = 0x6a09e667f3bcc909ULL;

std::uint64_t random_value() {
    random_state ^= random_state << 7;
    random_state ^= random_state >> 9;
    return random_state;
}

int random_int(int lower, int upper) {
    return lower + int(random_value() % std::uint64_t(upper - lower + 1));
}

void check_queries(
    const Matrix& matrix,
    const std::vector<int>& values,
    const std::vector<long long>& weights
) {
    assert(values.size() == weights.size());
    assert(matrix.size() == int(values.size()));
    assert(matrix.empty() == values.empty());
    for (int i = 0; i < int(values.size()); i++) {
        assert(matrix.access(i) == values[i]);
        assert(matrix[i] == values[i]);
        assert(matrix.weight(i) == weights[i]);
        std::pair<int, long long> expected(values[i], weights[i]);
        assert(matrix.get(i) == expected);
    }

    int left = int(random_value() % (values.size() + 1));
    int right = int(random_value() % (values.size() + 1));
    if (right < left) std::swap(left, right);
    int lower = random_int(-25, 25);
    int upper = random_int(-25, 25);
    if (upper < lower) std::swap(lower, upper);

    long long total = 0;
    long long below = 0;
    long long between = 0;
    int frequency = 0;
    std::vector<std::pair<int, int>> order;
    for (int i = left; i < right; i++) {
        total += weights[i];
        if (values[i] < upper) below += weights[i];
        if (lower <= values[i] && values[i] < upper) {
            between += weights[i];
            frequency++;
        }
        order.emplace_back(values[i], i);
    }
    std::stable_sort(
        order.begin(),
        order.end(),
        [](const auto& first, const auto& second) {
            return first.first < second.first;
        }
    );

    assert(matrix.range_sum(left, right) == total);
    assert(matrix.range_sum(left, right, upper) == below);
    assert(matrix.range_sum(left, right, lower, upper) == between);
    assert(matrix.range_freq(left, right, lower, upper) == frequency);

    long long smallest_sum = 0;
    long long largest_sum = 0;
    for (int k = 0; k <= int(order.size()); k++) {
        assert(matrix.sum_k_smallest(left, right, k) == smallest_sum);
        assert(matrix.sum_k_largest(left, right, k) == largest_sum);
        if (k < int(order.size())) {
            smallest_sum += weights[order[k].second];
            largest_sum += weights[order[order.size() - 1 - k].second];
        }
    }

    if (!order.empty()) {
        int k = int(random_value() % order.size());
        assert(matrix.kth_smallest(left, right, k) == order[k].first);
        assert(
            matrix.kth_largest(left, right, k) ==
            order[order.size() - 1 - k].first
        );
    }

    int value = random_int(-20, 20);
    assert(
        matrix.rank(value, left, right) ==
        int(std::count(values.begin() + left, values.begin() + right, value))
    );
    assert(
        matrix.rank(value, right) ==
        int(std::count(values.begin(), values.begin() + right, value))
    );

    std::optional<int> previous;
    std::optional<int> next;
    for (int i = left; i < right; i++) {
        if (values[i] < upper &&
            (!previous.has_value() || previous.value() < values[i])) {
            previous = values[i];
        }
        if (lower <= values[i] &&
            (!next.has_value() || values[i] < next.value())) {
            next = values[i];
        }
    }
    assert(matrix.prev_value(left, right, upper) == previous);
    assert(matrix.next_value(left, right, lower) == next);
}

void test_randomized() {
    std::vector<int> default_values;
    default_values.push_back(std::numeric_limits<int>::min());
    default_values.push_back(-2);
    default_values.push_back(0);
    default_values.push_back(std::numeric_limits<int>::max());
    m1une::ds::DynamicWaveletMatrixSum<int, long long> default_matrix(
        default_values
    );
    assert(
        default_matrix.range_sum(0, 4) ==
        static_cast<long long>(std::numeric_limits<int>::min()) - 2 +
            std::numeric_limits<int>::max()
    );
    default_matrix.set(1, 10);
    assert(default_matrix.weight(1) == 10);
    default_matrix.insert(2, -7);
    assert(default_matrix.weight(2) == -7);

    std::vector<unsigned long long> unsigned_values;
    unsigned_values.push_back(0);
    unsigned_values.push_back(std::numeric_limits<unsigned long long>::max());
    std::vector<long long> unsigned_weights;
    unsigned_weights.push_back(3);
    unsigned_weights.push_back(4);
    m1une::ds::DynamicWaveletMatrixSum<unsigned long long, long long>
        unsigned_matrix(unsigned_values, unsigned_weights);
    assert(unsigned_matrix.kth_largest(0, 2, 0) == unsigned_values[1]);
    assert(unsigned_matrix.range_sum(0, 2) == 7);
    unsigned_matrix.clear();
    assert(unsigned_matrix.empty());

    for (int trial = 0; trial < 80; trial++) {
        int initial_size = random_int(0, 60);
        std::vector<int> values(initial_size);
        std::vector<long long> weights(initial_size);
        for (int i = 0; i < initial_size; i++) {
            values[i] = random_int(-20, 20);
            weights[i] = random_int(-50, 50);
        }
        Matrix matrix(values, weights);

        for (int operation = 0; operation < 220; operation++) {
            int type = int(random_value() % 8);
            if (values.empty()) type = 0;
            if (type == 0) {
                int position = int(random_value() % (values.size() + 1));
                int value = random_int(-20, 20);
                long long weight = random_int(-50, 50);
                values.insert(values.begin() + position, value);
                weights.insert(weights.begin() + position, weight);
                matrix.insert(position, value, weight);
            } else if (type == 1) {
                int position = int(random_value() % values.size());
                std::pair<int, long long> expected(
                    values[position],
                    weights[position]
                );
                values.erase(values.begin() + position);
                weights.erase(weights.begin() + position);
                assert(matrix.erase(position) == expected);
            } else if (type == 2) {
                int position = int(random_value() % values.size());
                int value = random_int(-20, 20);
                long long weight = random_int(-50, 50);
                values[position] = value;
                weights[position] = weight;
                matrix.set(position, value, weight);
            } else if (type == 3) {
                int position = int(random_value() % values.size());
                int value = random_int(-20, 20);
                values[position] = value;
                matrix.set_value(position, value);
            } else if (type == 4) {
                int position = int(random_value() % values.size());
                long long weight = random_int(-50, 50);
                weights[position] = weight;
                matrix.set_weight(position, weight);
            } else if (type == 5) {
                int position = int(random_value() % values.size());
                long long delta = random_int(-20, 20);
                weights[position] += delta;
                matrix.add_weight(position, delta);
            } else if (type == 6) {
                int value = random_int(-20, 20);
                values.push_back(value);
                weights.push_back(value);
                matrix.push_back(value);
            } else {
                int position = int(random_value() % values.size());
                int value = random_int(-20, 20);
                values[position] = value;
                weights[position] = value;
                matrix.set(position, value);
            }
            check_queries(matrix, values, weights);
        }
    }

    std::vector<int> values(700);
    std::vector<long long> weights(700);
    for (int i = 0; i < int(values.size()); i++) {
        values[i] = random_int(-100, 100);
        weights[i] = random_int(-1000, 1000);
    }
    Matrix matrix(values, weights);
    for (int operation = 0; operation < 2500; operation++) {
        int from = int(random_value() % values.size());
        int to = int(random_value() % values.size());
        int value = values[from];
        long long weight = weights[from];
        values.erase(values.begin() + from);
        weights.erase(weights.begin() + from);
        std::pair<int, long long> expected(value, weight);
        assert(matrix.erase(from) == expected);
        values.insert(values.begin() + to, value);
        weights.insert(weights.begin() + to, weight);
        matrix.insert(to, value, weight);
        if (operation % 25 == 0) check_queries(matrix, values, weights);
    }
}

void test_boundary_search() {
    {
        std::vector<int> equal_values(700, 7);
        std::vector<long long> equal_weights(700);
        for (int i = 0; i < int(equal_weights.size()); i++) {
            equal_weights[i] = i % 17 + 1;
        }
        Matrix equal_matrix(equal_values, equal_weights);
        for (int operation = 0; operation < 300; operation++) {
            int type = int(random_value() % 3);
            if (type == 0) {
                int position =
                    int(random_value() % (equal_values.size() + 1));
                long long weight = random_int(1, 20);
                equal_values.insert(equal_values.begin() + position, 7);
                equal_weights.insert(
                    equal_weights.begin() + position,
                    weight
                );
                equal_matrix.insert(position, 7, weight);
            } else if (type == 1 && equal_values.size() > 300) {
                int position = int(random_value() % equal_values.size());
                equal_values.erase(equal_values.begin() + position);
                equal_weights.erase(equal_weights.begin() + position);
                equal_matrix.erase(position);
            } else {
                int position = int(random_value() % equal_values.size());
                equal_weights[position] = random_int(1, 20);
                equal_matrix.set_weight(position, equal_weights[position]);
            }

            int left = int(random_value() % (equal_values.size() + 1));
            int right = int(random_value() % (equal_values.size() + 1));
            if (right < left) std::swap(left, right);
            long long total = 0;
            for (int i = left; i < right; i++) total += equal_weights[i];
            long long limit = static_cast<long long>(
                random_value() % std::uint64_t(total + 21)
            );
            auto predicate = [limit](long long sum) {
                return sum <= limit;
            };

            long long sum = 0;
            int smallest = 0;
            while (left + smallest < right &&
                   sum + equal_weights[left + smallest] <= limit) {
                sum += equal_weights[left + smallest];
                smallest++;
            }
            assert(
                equal_matrix.max_count_smallest(
                    left,
                    right,
                    predicate
                ) == smallest
            );

            sum = 0;
            int largest = 0;
            while (left + largest < right &&
                   sum + equal_weights[right - 1 - largest] <= limit) {
                sum += equal_weights[right - 1 - largest];
                largest++;
            }
            assert(
                equal_matrix.max_count_largest(left, right, predicate) ==
                largest
            );
        }
    }

    std::vector<int> values(700);
    std::vector<long long> weights(700);
    for (int i = 0; i < int(values.size()); i++) {
        values[i] = random_int(-30, 30);
        weights[i] = random_int(1, 20);
    }
    Matrix matrix(values, weights);

    for (int operation = 0; operation < 700; operation++) {
        int type = int(random_value() % 4);
        if (type == 0) {
            int position = int(random_value() % (values.size() + 1));
            int value = random_int(-30, 30);
            long long weight = random_int(1, 20);
            values.insert(values.begin() + position, value);
            weights.insert(weights.begin() + position, weight);
            matrix.insert(position, value, weight);
        } else if (type == 1 && values.size() > 300) {
            int position = int(random_value() % values.size());
            values.erase(values.begin() + position);
            weights.erase(weights.begin() + position);
            matrix.erase(position);
        } else if (type == 2) {
            int position = int(random_value() % values.size());
            values[position] = random_int(-30, 30);
            matrix.set_value(position, values[position]);
        } else {
            int position = int(random_value() % values.size());
            weights[position] = random_int(1, 20);
            matrix.set_weight(position, weights[position]);
        }

        int left = int(random_value() % (values.size() + 1));
        int right = int(random_value() % (values.size() + 1));
        if (right < left) std::swap(left, right);
        std::vector<std::pair<int, int>> order;
        long long total = 0;
        for (int i = left; i < right; i++) {
            order.emplace_back(values[i], i);
            total += weights[i];
        }
        std::stable_sort(
            order.begin(),
            order.end(),
            [](const auto& first, const auto& second) {
                return first.first < second.first;
            }
        );
        long long limit = static_cast<long long>(
            random_value() % std::uint64_t(total + 21)
        );
        auto predicate = [limit](long long sum) { return sum <= limit; };

        long long sum = 0;
        int smallest = 0;
        while (smallest < int(order.size()) &&
               sum + weights[order[smallest].second] <= limit) {
            sum += weights[order[smallest].second];
            smallest++;
        }
        assert(
            matrix.max_count_smallest(left, right, predicate) == smallest
        );

        sum = 0;
        int largest = 0;
        while (largest < int(order.size()) &&
               sum + weights[order[order.size() - 1 - largest].second] <=
                   limit) {
            sum += weights[order[order.size() - 1 - largest].second];
            largest++;
        }
        assert(matrix.max_count_largest(left, right, predicate) == largest);
    }
}

struct Point {
    int x = 0;
    int y = 0;
    long long weight = 0;
    int query_index = -1;
};

struct Query {
    int type = 0;
    int first = 0;
    int second = 0;
    int third = 0;
    int fourth = 0;
    long long weight = 0;
    int position = -1;
};

}  // namespace

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

    test_randomized();
    test_boundary_search();

    int point_count = 0;
    int query_count = 0;
    fast_input >> point_count >> query_count;
    std::vector<Point> points(point_count);
    for (Point& point : points) {
        fast_input >> point.x >> point.y >> point.weight;
    }

    std::vector<Query> queries(query_count);
    for (int query_index = 0; query_index < query_count; query_index++) {
        Query& query = queries[query_index];
        fast_input >> query.type;
        if (query.type == 0) {
            fast_input >> query.first >> query.second >> query.weight;
            Point point;
            point.x = query.first;
            point.y = query.second;
            point.query_index = query_index;
            points.push_back(point);
        } else {
            fast_input >> query.first >> query.second >> query.third >>
                query.fourth;
        }
    }

    std::stable_sort(
        points.begin(),
        points.end(),
        [](const Point& first, const Point& second) {
            return first.x < second.x;
        }
    );
    std::vector<int> xs(points.size());
    std::vector<int> ys(points.size());
    std::vector<long long> weights(points.size());
    for (int i = 0; i < int(points.size()); i++) {
        xs[i] = points[i].x;
        ys[i] = points[i].y;
        weights[i] = points[i].weight;
        if (points[i].query_index != -1) {
            queries[points[i].query_index].position = i;
        }
    }

    Matrix matrix(ys, weights);
    for (const Query& query : queries) {
        if (query.type == 0) {
            matrix.set_weight(query.position, query.weight);
        } else {
            int left = int(std::lower_bound(
                               xs.begin(),
                               xs.end(),
                               query.first
                           ) -
                           xs.begin());
            int right = int(std::lower_bound(
                                xs.begin(),
                                xs.end(),
                                query.third
                            ) -
                            xs.begin());
            fast_output << matrix.range_sum(
                               left,
                               right,
                               query.second,
                               query.fourth
                           )
                        << '\n';
        }
    }
}

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