DynamicLazySegmentTree(動的遅延セグメント木)
(data-struct/segment/DynamicLazySegmentTree.hpp)

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• Last update: 2024-05-12 17:35:55+09:00
• Include: #include "data-struct/segment/DynamicLazySegmentTree.hpp"

概要

初期値以外の値を持っているか、作用素がかかっている要素だけ持つセグメント木。座標圧縮して通常の遅延セグメント木を使うのでもいいが、オンラインでもできることに利点がある。あと楽。

また、 merge や split も実装したが未 verify。

• コンストラクタ
  • DynamicLazySegmentTree() : 長さ $inf$ に初期化する。 $\Theta(\log n)$ 。
  • DynamicLazySegmentTree(ll n) : 長さ n で初期化する。初期値は e 。 $\Theta(\log n)$ 。
  • DynamicLazySegmentTree(ll n, T v) : 長さ n で初期化する。初期値は v 。 $\Theta(\log n)$ 。
  • void init(ll n, T v) : 長さ n で初期化する。初期値は v 。 $\Theta(N)$ 。
• 取得クエリ
  • T prod(ll l, ll r) : op(a[l], a[l+1], ..., a[r-1]) を返す。 $\Theta(\log N)$ 。
  • T get(ll k) : a[k] を返す。 $\Theta(\log N)$ 。
  • T all_prod() : op(a[0], a[1], ..., a[N-1]) を返す。 $\Theta(1)$ 。
• 更新クエリ
  • void set(ll k, T x) : a[k] に x を代入する。 $\Theta(\log N)$ 。
  • void apply(ll k, U f) : a[k] に mp(f, a[k]) を代入する。 $\Theta(\log N)$ 。
  • void update(ll k, T upd(T)) : a[k] に upd(a[k]) を代入する。 $\Theta(\log N)$ 。
  • void apply(ll l, ll r, U f) : a[l], a[l+1], ..., a[r-1] に mp(f, a[l]), mp(f, a[l+1]), ..., mp(f, a[r-1]) を代入する。 $\Theta(\log N)$ 。
• セグメント木上の二分探索
  • ll max_right(ll l, bool f(T)) :
    [l, r) に対して f が true を返すような最大の r を返す。f(e) = true である必要がある。 $\Theta(\log N)$ 。
    厳密には、以下の条件を共に満たす r (のうち1つ)を返す。
    • r = l または f(op(a[l], a[l+1], ..., a[r-1])) = true
    • r = n または f(op(a[l], a[l+1], ..., a[r])) = false
  • ll min_left(ll r, bool f(T)) :
    [l, r) に対して f が true を返すような最小の l を返す。f(e) = true である必要がある。 $\Theta(\log N)$ 。
    厳密には、以下の条件を共に満たす l (のうち1つ)を返す。
    • l = r または f(op(a[l], a[l+1], ..., a[r-1])) = true
    • l = 0 または f(op(a[l-1], a[l], ..., a[r-1])) = false

Depends on

• other/monoid.hpp
• other/template.hpp
• template/alias.hpp
• template/bitop.hpp
• template/func.hpp
• template/in.hpp
• template/macros.hpp
• template/out.hpp
• template/type_traits.hpp
• template/util.hpp

Verified with

• test/aoj/DSL/DSL_2_F-RUQRMQ-DynamicLazySeg.test.cpp
• test/aoj/DSL/DSL_2_G-RAQRSQ-DynamicLazySeg.test.cpp

Code

#pragma once

#include "../../other/template.hpp"
#include "../../other/monoid.hpp"

template<class A> class DynamicLazySegmentTree {
private:
    using M = typename A::M;
    using E = typename A::E;
    using T = typename M::value_type;
    using U = typename E::value_type;
    struct node;
    using node_ptr = std::unique_ptr<node>;
    struct node {
        T val;
        U lazy;
        bool lazyflag;
        node_ptr l, r;
        node(const T& v) : val(v), lazyflag(false), l(nullptr), r(nullptr) {}
        node(const T& v, const U& x)
            : val(v), lazy(x), lazyflag(true), l(nullptr), r(nullptr) {}
    };
    node_ptr& get_l(const node_ptr& nd) const {
        if (nd->l == nullptr) nd->l = std::make_unique<node>(M::id());
        return nd->l;
    }
    node_ptr& get_r(const node_ptr& nd) const {
        if (nd->r == nullptr) nd->r = std::make_unique<node>(M::id());
        return nd->r;
    }

    template<bool AlwaysTrue = true,
             typename std::enable_if<!Monoid::has_mul_op<A>::value &&
                                     AlwaysTrue>::type* = nullptr>
    static inline T Aop(const U& a, const T& b, ll) {
        return A::op(a, b);
    }
    template<bool AlwaysTrue = true,
             typename std::enable_if<Monoid::has_mul_op<A>::value &&
                                     AlwaysTrue>::type* = nullptr>
    static inline T Aop(const U& a, const T& b, ll c) {
        return A::mul_op(a, c, b);
    }

    void all_apply(node_ptr& nd, int t, const U& x, ll d) {
        nd->val = Aop(x, nd->val, d);
        if (t != 0) {
            if (nd->lazyflag) {
                nd->lazy = E::op(nd->lazy, x);
            }
            else {
                nd->lazy = x;
                nd->lazyflag = true;
            }
        }
    }
    void eval(node_ptr& nd, ll a, ll b, int t) {
        if (nd->lazyflag) {
            ll m = (a + b) >> 1;
            all_apply(get_l(nd), t - 1, nd->lazy, m - a);
            all_apply(get_r(nd), t - 1, nd->lazy, b - m);
            nd->lazyflag = false;
        }
    }
    ll ori, h, n;
    std::vector<T> iv, iv2;
    node_ptr root;
    template<class Upd>
    void update(node_ptr& nd, ll a, ll b, int t, ll k, const Upd& upd) {
        if (a + 1 == b) {
            nd->val = upd(nd->val);
            return;
        }
        eval(nd, a, b, t);
        ll m = (a + b) >> 1;
        if (k < m) update(get_l(nd), a, m, t - 1, k, upd);
        else update(get_r(nd), m, b, t - 1, k, upd);
        nd->val =
            M::op(nd->l ? nd->l->val : M::id(), nd->r ? nd->r->val : M::id());
    }
    T prod(node_ptr& nd, ll a, ll b, int t, ll l, ll r) {
        if (nd == nullptr) return M::id();
        if (r <= a || b <= l) return M::id();
        if (l <= a && b <= r) return nd->val;
        eval(nd, a, b, t);
        ll m = (a + b) >> 1;
        return M::op(prod(nd->l, a, m, t - 1, l, r),
                     prod(nd->r, m, b, t - 1, l, r));
    }
    void apply(node_ptr& nd, ll a, ll b, int t, ll l, ll r, const U& x) {
        if (r <= a || b <= l) return;
        if (l <= a && b <= r) {
            all_apply(nd, t, x, b - a);
            return;
        }
        eval(nd, a, b, t);
        ll m = (a + b) >> 1;
        apply(get_l(nd), a, m, t - 1, l, r, x);
        apply(get_r(nd), m, b, t - 1, l, r, x);
        nd->val =
            M::op(nd->l ? nd->l->val : M::id(), nd->r ? nd->r->val : M::id());
    }
    template<class Cond>
    ll max_right(node_ptr& nd, ll a, ll b, int t, ll l, const Cond& cond,
                 T& sm) {
        if (b <= l || nd == nullptr) return n;
        if (l <= a && cond(M::op(sm, nd->val))) {
            sm = M::op(sm, nd->val);
            return n;
        }
        if (a + 1 == b) return a;
        eval(nd, a, b, t);
        ll m = (a + b) >> 1;
        ll res = max_right(nd->l, a, m, t - 1, l, cond, sm);
        if (res != n) return res;
        return max_right(nd->r, m, b, t - 1, l, cond, sm);
    }
    template<class Cond>
    ll min_left(node_ptr& nd, ll a, ll b, int t, ll r, const Cond& cond,
                T& sm) {
        if (r <= a || nd == nullptr) return 0;
        if (b <= r && cond(M::op(nd->val, sm))) {
            sm = M::op(nd->val, sm);
            return 0;
        }
        if (a + 1 == b) return b;
        eval(nd, a, b, t);
        ll m = (a + b) >> 1;
        ll res = min_left(nd->r, m, b, t - 1, r, cond, sm);
        if (res != 0) return res;
        return min_left(nd->l, a, m, t - 1, r, cond, sm);
    }
    void reset(node_ptr& nd, ll a, ll b, int t, ll l, ll r) {
        if (nd == nullptr) return;
        if (r <= a || b <= l) return;
        if (l <= a && b <= r) {
            if (nd == root) nd = std::make_unique<node>(M::id());
            else nd.reset();
            return;
        }
        ll m = (a + b) >> 1;
        reset(nd->l, a, m, t - 1, l, r);
        reset(nd->r, m, b, t - 1, l, r);
        nd->val =
            M::op(nd->l ? nd->l->val : M::id(), nd->r ? nd->r->val : M::id());
    }
    void merge(node_ptr& nd, ll a, ll b, int t, node_ptr& other) {
        if (nd == nullptr) {
            nd = std::move(other);
            return;
        }
        if (other == nullptr) return;
        if (nd->l == nullptr && nd->r == nullptr) {
            nd->val = M::op(nd->val, other->val);
            return;
        }
        eval(nd, a, b, t);
        eval(other, a, b, t);
        ll m = (a + b) >> 1;
        merge(nd->l, a, m, t - 1, other->l);
        merge(nd->r, m, b, t - 1, other->r);
        nd->val =
            M::op(nd->l ? nd->l->val : M::id(), nd->r ? nd->r->val : M::id());
    }
    void split(node_ptr& nd, ll a, ll b, int t, node_ptr& other, ll k) {
        if (nd == nullptr) return;
        if (a >= k) {
            other = std::move(nd);
            nd = nullptr;
            return;
        }
        if (b <= k) return;
        other = std::make_unique<node>(M::id());
        eval(nd, a, b, t);
        eval(other, a, b, t);
        ll m = (a + b) >> 1;
        split(nd->l, a, m, t - 1, other->l, k);
        split(nd->r, m, b, t - 1, other->r, k);
        if (nd->l == nullptr && nd->r == nullptr) nd = nullptr;
        else {
            nd->val = M::op(nd->l ? nd->l->val : M::id(),
                            nd->r ? nd->r->val : M::id());
        }
        if (other->l == nullptr && other->r == nullptr) other = nullptr;
        else {
            other->val = M::op(other->l ? other->l->val : M::id(),
                               other->r ? other->r->val : M::id());
        }
    }
    void init_copy(node_ptr& nd, const node_ptr& src) {
        if (src == nullptr) return;
        if (src->lazyflag) nd = std::make_unique<node>(src->val, src->lazy);
        else nd = std::make_unique<node>(src->val);
        init_copy(nd->l, src->l);
        init_copy(nd->r, src->r);
    }

public:
    DynamicLazySegmentTree() : DynamicLazySegmentTree(inf) {}
    DynamicLazySegmentTree(ll n_) { init(n_); }
    DynamicLazySegmentTree(ll n_, const T& v) { init(n_); }
    DynamicLazySegmentTree(const DynamicLazySegmentTree& other)
        : n(other.n), h(other.h), ori(other.ori), iv(other.iv), iv2(other.iv2),
          root(std::make_unique<node>(other.root->val)) {
        init_copy(root, other.root);
    }
    DynamicLazySegmentTree(DynamicLazySegmentTree&& other) = default;
    DynamicLazySegmentTree& operator=(const DynamicLazySegmentTree& other) {
        if (this == &other) return *this;
        return (*this) = DynamicLazySegmentTree(other);
    }
    DynamicLazySegmentTree& operator=(DynamicLazySegmentTree&& other) = default;
    void init(ll n_) {
        ori = n_;
        h = bitop::ceil_log2(ori);
        n = 1ull << h;
        root = std::make_unique<node>(M::id());
    }
    template<class Upd> void update(ll k, const Upd& upd) {
        assert(0 <= k && k < ori);
        update(root, 0, n, h, k, upd);
    }
    void set(ll k, T x) {
        update(k, [&](const T&) -> T { return x; });
    }
    void apply(ll k, const U& x) {
        update(k, [&](const T& a) -> T { return A::op(x, a); });
    }
    void apply(ll l, ll r, const U& x) {
        assert(0 <= l && l <= r && r <= ori);
        apply(root, 0, n, h, l, r, x);
    }
    void all_apply(const U& x) { apply(root, 0, n, h, 0, n, x); }
    T prod(ll l, ll r) {
        assert(0 <= l && l <= r && r <= ori);
        return prod(root, 0, n, h, l, r);
    }
    T all_prod() const { return root->val; }
    T get(ll k) { return prod(k, k + 1); }
    template<class Cond> ll max_right(ll l, const Cond& cond) {
        assert(0 <= l && l <= ori);
        if (l == n) return n;
        T sm = M::id();
        assert(cond(sm));
        return std::min(max_right(root, 0, n, h, l, cond, sm), ori);
    }
    template<class Cond> ll min_left(ll r, const Cond& cond) {
        assert(0 <= r && r <= ori);
        if (0 == r) return 0;
        T sm = M::id();
        assert(cond(sm));
        return min_left(root, 0, n, h, r, cond, sm);
    }
    void reset(ll l, ll r) { reset(root, 0, n, h, l, r); }
    void reset(ll k) { reset(root, 0, n, h, k, k + 1); }
    DynamicLazySegmentTree& merge(DynamicLazySegmentTree&& other) {
        assert(ori == other.ori);
        merge(root, 0, n, h, other.root);
        other.root = std::make_unique<node>(M::id());
        return *this;
    }
    friend DynamicLazySegmentTree merge(DynamicLazySegmentTree&& a,
                                        DynamicLazySegmentTree&& b) {
        return std::move(a.merge(std::move(b)));
    }
    friend std::pair<DynamicLazySegmentTree, DynamicLazySegmentTree>
    split(DynamicLazySegmentTree&& a, ll k) {
        assert(0 <= k && k <= a.ori);
        DynamicLazySegmentTree b(a.ori);
        a.split(a.root, 0, a.n, a.h, b.root, k);
        return {std::move(a), std::move(b)};
    }
};

/**
 * @brief DynamicLazySegmentTree(動的遅延セグメント木)
 * @docs docs/data-struct/segment/DynamicLazySegmentTree.md
 */

#line 2 "data-struct/segment/DynamicLazySegmentTree.hpp"

#line 2 "other/template.hpp"

#include <bits/stdc++.h>
#line 2 "template/macros.hpp"

#line 4 "template/macros.hpp"

#ifndef __COUNTER__
#define __COUNTER__ __LINE__
#endif

#define OVERLOAD5(a, b, c, d, e, ...) e
#define REP1_0(b, c) REP1_1(b, c)
#define REP1_1(b, c)                                                           \
    for (ll REP_COUNTER_##c = 0; REP_COUNTER_##c < (ll)(b); ++REP_COUNTER_##c)
#define REP1(b) REP1_0(b, __COUNTER__)
#define REP2(i, b) for (ll i = 0; i < (ll)(b); ++i)
#define REP3(i, a, b) for (ll i = (ll)(a); i < (ll)(b); ++i)
#define REP4(i, a, b, c) for (ll i = (ll)(a); i < (ll)(b); i += (ll)(c))
#define rep(...) OVERLOAD5(__VA_ARGS__, REP4, REP3, REP2, REP1)(__VA_ARGS__)
#define RREP2(i, a) for (ll i = (ll)(a)-1; i >= 0; --i)
#define RREP3(i, a, b) for (ll i = (ll)(a)-1; i >= (ll)(b); --i)
#define RREP4(i, a, b, c) for (ll i = (ll)(a)-1; i >= (ll)(b); i -= (ll)(c))
#define rrep(...) OVERLOAD5(__VA_ARGS__, RREP4, RREP3, RREP2)(__VA_ARGS__)
#define REPS2(i, b) for (ll i = 1; i <= (ll)(b); ++i)
#define REPS3(i, a, b) for (ll i = (ll)(a) + 1; i <= (ll)(b); ++i)
#define REPS4(i, a, b, c) for (ll i = (ll)(a) + 1; i <= (ll)(b); i += (ll)(c))
#define reps(...) OVERLOAD5(__VA_ARGS__, REPS4, REPS3, REPS2)(__VA_ARGS__)
#define RREPS2(i, a) for (ll i = (ll)(a); i > 0; --i)
#define RREPS3(i, a, b) for (ll i = (ll)(a); i > (ll)(b); --i)
#define RREPS4(i, a, b, c) for (ll i = (ll)(a); i > (ll)(b); i -= (ll)(c))
#define rreps(...) OVERLOAD5(__VA_ARGS__, RREPS4, RREPS3, RREPS2)(__VA_ARGS__)

#define each_for(...) for (auto&& __VA_ARGS__)
#define each_const(...) for (const auto& __VA_ARGS__)

#define all(v) std::begin(v), std::end(v)
#define rall(v) std::rbegin(v), std::rend(v)

#if __cpp_if_constexpr >= 201606L
#define IF_CONSTEXPR constexpr
#else
#define IF_CONSTEXPR
#endif

#define IO_BUFFER_SIZE (1 << 17)
#line 2 "template/alias.hpp"

#line 4 "template/alias.hpp"

using ll = long long;
using uint = unsigned int;
using ull = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
using ld = long double;
using PLL = std::pair<ll, ll>;
template<class T>
using prique = std::priority_queue<T, std::vector<T>, std::greater<T>>;

template<class T> struct infinity {
    static constexpr T value = std::numeric_limits<T>::max() / 2;
    static constexpr T mvalue = std::numeric_limits<T>::lowest() / 2;
    static constexpr T max = std::numeric_limits<T>::max();
    static constexpr T min = std::numeric_limits<T>::lowest();
};

#if __cplusplus <= 201402L
template<class T> constexpr T infinity<T>::value;
template<class T> constexpr T infinity<T>::mvalue;
template<class T> constexpr T infinity<T>::max;
template<class T> constexpr T infinity<T>::min;
#endif

#if __cpp_variable_templates >= 201304L
template<class T> constexpr T INF = infinity<T>::value;
#endif

constexpr ll inf = infinity<ll>::value;
constexpr ld EPS = 1e-8;
constexpr ld PI = 3.1415926535897932384626;
#line 2 "template/type_traits.hpp"

#line 5 "template/type_traits.hpp"

template<class T, class... Args> struct function_traits_impl {
    using result_type = T;
    template<std::size_t idx>
    using argument_type =
        typename std::tuple_element<idx, std::tuple<Args...>>::type;
    using argument_tuple = std::tuple<Args...>;
    static constexpr std::size_t arg_size() { return sizeof...(Args); }
};

template<class> struct function_traits_helper;

template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...)> {
    using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...)&> {
    using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) const> {
    using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) const&> {
    using type = function_traits_impl<Res, Args...>;
};

#if __cpp_noexcept_function_type >= 201510L
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) noexcept> {
    using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...)& noexcept> {
    using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) const noexcept> {
    using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) const& noexcept> {
    using type = function_traits_impl<Res, Args...>;
};
#endif

template<class F>
using function_traits = typename function_traits_helper<
    decltype(&std::remove_reference<F>::type::operator())>::type;

template<class F>
using function_result_type = typename function_traits<F>::result_type;
template<class F, std::size_t idx>
using function_argument_type =
    typename function_traits<F>::template argument_type<idx>;
template<class F>
using function_argument_tuple = typename function_traits<F>::argument_tuple;

template<class T>
using is_signed_int =
    std::integral_constant<bool, (std::is_integral<T>::value &&
                                  std::is_signed<T>::value) ||
                                     std::is_same<T, i128>::value>;
template<class T>
using is_unsigned_int =
    std::integral_constant<bool, (std::is_integral<T>::value &&
                                  std::is_unsigned<T>::value) ||
                                     std::is_same<T, u128>::value>;
template<class T>
using is_int = std::integral_constant<bool, is_signed_int<T>::value ||
                                                is_unsigned_int<T>::value>;
template<class T>
using make_signed_int = typename std::conditional<
    std::is_same<T, i128>::value || std::is_same<T, u128>::value,
    std::common_type<i128>, std::make_signed<T>>::type;
template<class T>
using make_unsigned_int = typename std::conditional<
    std::is_same<T, i128>::value || std::is_same<T, u128>::value,
    std::common_type<u128>, std::make_unsigned<T>>::type;


template<class T, class = void> struct is_range : std::false_type {};
template<class T>
struct is_range<
    T,
    decltype(all(std::declval<typename std::add_lvalue_reference<T>::type>()),
             (void)0)> : std::true_type {};

template<class T, bool = is_range<T>::value>
struct range_rank : std::integral_constant<std::size_t, 0> {};
template<class T>
struct range_rank<T, true>
    : std::integral_constant<std::size_t,
                             range_rank<typename T::value_type>::value + 1> {};

template<std::size_t size> struct int_least {
    static_assert(size <= 128, "size must be less than or equal to 128");

    using type = typename std::conditional<
        size <= 8, std::int_least8_t,
        typename std::conditional<
            size <= 16, std::int_least16_t,
            typename std::conditional<
                size <= 32, std::int_least32_t,
                typename std::conditional<size <= 64, std::int_least64_t,
                                          i128>::type>::type>::type>::type;
};

template<std::size_t size> using int_least_t = typename int_least<size>::type;

template<std::size_t size> struct uint_least {
    static_assert(size <= 128, "size must be less than or equal to 128");

    using type = typename std::conditional<
        size <= 8, std::uint_least8_t,
        typename std::conditional<
            size <= 16, std::uint_least16_t,
            typename std::conditional<
                size <= 32, std::uint_least32_t,
                typename std::conditional<size <= 64, std::uint_least64_t,
                                          u128>::type>::type>::type>::type;
};

template<std::size_t size> using uint_least_t = typename uint_least<size>::type;

template<class T>
using double_size_int = int_least<std::numeric_limits<T>::digits * 2 + 1>;
template<class T> using double_size_int_t = typename double_size_int<T>::type;
template<class T>
using double_size_uint = uint_least<std::numeric_limits<T>::digits * 2>;
template<class T> using double_size_uint_t = typename double_size_uint<T>::type;

template<class T>
using double_size =
    typename std::conditional<is_signed_int<T>::value, double_size_int<T>,
                              double_size_uint<T>>::type;
template<class T> using double_size_t = typename double_size<T>::type;
#line 2 "template/in.hpp"

#line 4 "template/in.hpp"
#include <unistd.h>
#line 8 "template/in.hpp"

template<std::size_t buf_size = IO_BUFFER_SIZE,
         std::size_t decimal_precision = 16>
class Scanner {
private:
    template<class, class = void> struct has_scan : std::false_type {};
    template<class T>
    struct has_scan<
        T, decltype(std::declval<T>().scan(std::declval<Scanner&>()), (void)0)>
        : std::true_type {};
    int fd;
    int idx, sz;
    bool state;
    std::array<char, IO_BUFFER_SIZE + 1> buffer;
    inline char cur() {
        if (idx == sz) load();
        if (idx == sz) {
            state = false;
            return '\0';
        }
        return buffer[idx];
    }
    inline void next() {
        if (idx == sz) load();
        if (idx == sz) return;
        ++idx;
    }

public:
    inline void load() {
        int len = sz - idx;
        if (idx < len) return;
        std::memcpy(buffer.begin(), buffer.begin() + idx, len);
        sz = len + read(fd, buffer.data() + len, buf_size - len);
        buffer[sz] = 0;
        idx = 0;
    }

    Scanner(int fd) : fd(fd), idx(0), sz(0), state(true) {}
    Scanner(FILE* fp) : fd(fileno(fp)), idx(0), sz(0), state(true) {}

    inline char scan_char() {
        if (idx == sz) load();
        return idx == sz ? '\0' : buffer[idx++];
    }

    Scanner ignore(int n = 1) {
        if (idx + n > sz) load();
        idx += n;
        return *this;
    }

    inline void discard_space() {
        if (idx == sz) load();
        while (('\t' <= buffer[idx] && buffer[idx] <= '\r') ||
               buffer[idx] == ' ') {
            if (++idx == sz) load();
        }
    }
    void scan(char& a) {
        discard_space();
        a = scan_char();
    }
    void scan(bool& a) {
        discard_space();
        a = scan_char() != '0';
    }
    void scan(std::string& a) {
        discard_space();
        a.clear();
        while (cur() != '\0' && (buffer[idx] < '\t' || '\r' < buffer[idx]) &&
               buffer[idx] != ' ') {
            a += scan_char();
        }
    }
    template<std::size_t len> void scan(std::bitset<len>& a) {
        discard_space();
        if (idx + len > sz) load();
        rrep (i, len) a[i] = buffer[idx++] != '0';
    }
    template<class T,
             typename std::enable_if<is_signed_int<T>::value &&
                                     !has_scan<T>::value>::type* = nullptr>
    void scan(T& a) {
        discard_space();
        if (buffer[idx] == '-') {
            ++idx;
            if (idx + 40 > sz &&
                (idx == sz || ('0' <= buffer[sz - 1] && buffer[sz - 1] <= '9')))
                load();
            a = 0;
            while ('0' <= buffer[idx] && buffer[idx] <= '9') {
                a = a * 10 - (buffer[idx++] - '0');
            }
        }
        else {
            if (idx + 40 > sz && '0' <= buffer[sz - 1] && buffer[sz - 1] <= '9')
                load();
            a = 0;
            while ('0' <= buffer[idx] && buffer[idx] <= '9') {
                a = a * 10 + (buffer[idx++] - '0');
            }
        }
    }
    template<class T,
             typename std::enable_if<is_unsigned_int<T>::value &&
                                     !has_scan<T>::value>::type* = nullptr>
    void scan(T& a) {
        discard_space();
        if (idx + 40 > sz && '0' <= buffer[sz - 1] && buffer[sz - 1] <= '9')
            load();
        a = 0;
        while ('0' <= buffer[idx] && buffer[idx] <= '9') {
            a = a * 10 + (buffer[idx++] - '0');
        }
    }
    template<class T,
             typename std::enable_if<std::is_floating_point<T>::value &&
                                     !has_scan<T>::value>::type* = nullptr>
    void scan(T& a) {
        discard_space();
        bool sgn = false;
        if (cur() == '-') {
            sgn = true;
            next();
        }
        a = 0;
        while ('0' <= cur() && cur() <= '9') {
            a = a * 10 + cur() - '0';
            next();
        }
        if (cur() == '.') {
            next();
            T n = 0, d = 1;
            for (int i = 0;
                 '0' <= cur() && cur() <= '9' && i < (int)decimal_precision;
                 ++i) {
                n = n * 10 + cur() - '0';
                d *= 10;
                next();
            }
            while ('0' <= cur() && cur() <= '9') next();
            a += n / d;
        }
        if (sgn) a = -a;
    }

private:
    template<std::size_t i, class... Args> void scan(std::tuple<Args...>& a) {
        if IF_CONSTEXPR (i < sizeof...(Args)) {
            scan(std::get<i>(a));
            scan<i + 1, Args...>(a);
        }
    }

public:
    template<class... Args> void scan(std::tuple<Args...>& a) {
        scan<0, Args...>(a);
    }
    template<class T, class U> void scan(std::pair<T, U>& a) {
        scan(a.first);
        scan(a.second);
    }
    template<class T,
             typename std::enable_if<is_range<T>::value &&
                                     !has_scan<T>::value>::type* = nullptr>
    void scan(T& a) {
        for (auto&& i : a) scan(i);
    }
    template<class T,
             typename std::enable_if<has_scan<T>::value>::type* = nullptr>
    void scan(T& a) {
        a.scan(*this);
    }

    void operator()() {}
    template<class Head, class... Args>
    void operator()(Head& head, Args&... args) {
        scan(head);
        operator()(args...);
    }

    template<class T> Scanner& operator>>(T& a) {
        scan(a);
        return *this;
    }

    explicit operator bool() const { return state; }

    friend Scanner& getline(Scanner& scan, std::string& a) {
        a.erase();
        char c;
        if ((c = scan.scan_char()) == '\n' || c == '\0') return scan;
        a += c;
        while ((c = scan.scan_char()) != '\n' && c != '\0') a += c;
        scan.state = true;
        return scan;
    }
};

Scanner<> scan(0);
#line 2 "template/out.hpp"

#line 8 "template/out.hpp"

struct NumberToString {
    char buf[10000][4];
    constexpr NumberToString() : buf{} {
        rep (i, 10000) {
            int n = i;
            rrep (j, 4) {
                buf[i][j] = (char)('0' + n % 10);
                n /= 10;
            }
        }
    }
} constexpr precalc_number_to_string;

template<std::size_t buf_size = IO_BUFFER_SIZE, bool debug = false>
class Printer {
private:
    template<class, bool = debug, class = void>
    struct has_print : std::false_type {};
    template<class T>
    struct has_print<T, false,
                     decltype(std::declval<T>().print(std::declval<Printer&>()),
                              (void)0)> : std::true_type {};
    template<class T>
    struct has_print<T, true,
                     decltype(std::declval<T>().debug(std::declval<Printer&>()),
                              (void)0)> : std::true_type {};
    int fd;
    std::size_t idx;
    std::array<char, buf_size> buffer;

    std::size_t decimal_precision;

public:
    inline void print_char(char c) {
#if SHIO_LOCAL
        buffer[idx++] = c;
        if (idx == buf_size) flush();
#else
        if IF_CONSTEXPR (!debug) {
            buffer[idx++] = c;
            if (idx == buf_size) flush();
        }
#endif
    }
    inline void flush() {
        idx = write(fd, buffer.begin(), idx);
        idx = 0;
    }

    Printer(int fd) : fd(fd), idx(0), decimal_precision(16) {}
    Printer(FILE* fp) : fd(fileno(fp)), idx(0), decimal_precision(16) {}
    ~Printer() { flush(); }

    void set_decimal_precision(std::size_t decimal_precision) {
        this->decimal_precision = decimal_precision;
    }

    void print(char c) {
        if IF_CONSTEXPR (debug) print_char('\'');
        print_char(c);
        if IF_CONSTEXPR (debug) print_char('\'');
    }
    void print(bool b) { print_char((char)(b + '0')); }
    void print(const char* a) {
        if IF_CONSTEXPR (debug) print_char('"');
        for (; *a != '\0'; ++a) print_char(*a);
        if IF_CONSTEXPR (debug) print_char('"');
    }
    template<std::size_t len> void print(const char (&a)[len]) {
        if IF_CONSTEXPR (debug) print_char('"');
        for (auto i : a) print_char(i);
        if IF_CONSTEXPR (debug) print_char('"');
    }
    void print(const std::string& a) {
        if IF_CONSTEXPR (debug) print_char('"');
        for (auto i : a) print_char(i);
        if IF_CONSTEXPR (debug) print_char('"');
    }
    template<std::size_t len> void print(const std::bitset<len>& a) {
        rrep (i, len) print_char((char)(a[i] + '0'));
    }
    template<class T,
             typename std::enable_if<is_int<T>::value &&
                                     !has_print<T>::value>::type* = nullptr>
    void print(T a) {
        if (!a) {
            print_char('0');
            return;
        }
        if IF_CONSTEXPR (is_signed_int<T>::value) {
            if (a < 0) {
                print_char('-');
                using U = typename make_unsigned_int<T>::type;
                print(static_cast<U>(-static_cast<U>(a)));
                return;
            }
        }
        if (idx + 40 >= buf_size) flush();
        static char s[40];
        int t = 40;
        while (a >= 10000) {
            int i = a % 10000;
            a /= 10000;
            t -= 4;
            std::memcpy(s + t, precalc_number_to_string.buf[i], 4);
        }
        if (a >= 1000) {
            std::memcpy(buffer.begin() + idx, precalc_number_to_string.buf[a],
                        4);
            idx += 4;
        }
        else if (a >= 100) {
            std::memcpy(buffer.begin() + idx,
                        precalc_number_to_string.buf[a] + 1, 3);
            idx += 3;
        }
        else if (a >= 10) {
            std::memcpy(buffer.begin() + idx,
                        precalc_number_to_string.buf[a] + 2, 2);
            idx += 2;
        }
        else {
            buffer[idx++] = '0' | a;
        }
        std::memcpy(buffer.begin() + idx, s + t, 40 - t);
        idx += 40 - t;
    }
    template<class T,
             typename std::enable_if<std::is_floating_point<T>::value &&
                                     !has_print<T>::value>::type* = nullptr>
    void print(T a) {
        if (a == std::numeric_limits<T>::infinity()) {
            print("inf");
            return;
        }
        if (a == -std::numeric_limits<T>::infinity()) {
            print("-inf");
            return;
        }
        if (std::isnan(a)) {
            print("nan");
            return;
        }
        if (a < 0) {
            print_char('-');
            a = -a;
        }
        T b = a;
        if (b < 1) {
            print_char('0');
        }
        else {
            std::string s;
            while (b >= 1) {
                s += (char)('0' + (int)std::fmod(b, 10.0));
                b /= 10;
            }
            for (auto i = s.rbegin(); i != s.rend(); ++i) print_char(*i);
        }
        print_char('.');
        rep (decimal_precision) {
            a *= 10;
            print_char((char)('0' + (int)std::fmod(a, 10.0)));
        }
    }

private:
    template<std::size_t i, class... Args>
    void print(const std::tuple<Args...>& a) {
        if IF_CONSTEXPR (i < sizeof...(Args)) {
            if IF_CONSTEXPR (debug) print_char(',');
            print_char(' ');
            print(std::get<i>(a));
            print<i + 1, Args...>(a);
        }
    }

public:
    template<class... Args> void print(const std::tuple<Args...>& a) {
        if IF_CONSTEXPR (debug) print_char('(');
        if IF_CONSTEXPR (sizeof...(Args) != 0) print(std::get<0>(a));
        print<1, Args...>(a);
        if IF_CONSTEXPR (debug) print_char(')');
    }
    template<class T, class U> void print(const std::pair<T, U>& a) {
        if IF_CONSTEXPR (debug) print_char('(');
        print(a.first);
        if IF_CONSTEXPR (debug) print_char(',');
        print_char(' ');
        print(a.second);
        if IF_CONSTEXPR (debug) print_char(')');
    }
    template<class T,
             typename std::enable_if<is_range<T>::value &&
                                     !has_print<T>::value>::type* = nullptr>
    void print(const T& a) {
        if IF_CONSTEXPR (debug) print_char('{');
        for (auto i = std::begin(a); i != std::end(a); ++i) {
            if (i != std::begin(a)) {
                if IF_CONSTEXPR (debug) print_char(',');
                print_char(' ');
            }
            print(*i);
        }
        if IF_CONSTEXPR (debug) print_char('}');
    }
    template<class T, typename std::enable_if<has_print<T>::value &&
                                              !debug>::type* = nullptr>
    void print(const T& a) {
        a.print(*this);
    }
    template<class T, typename std::enable_if<has_print<T>::value &&
                                              debug>::type* = nullptr>
    void print(const T& a) {
        a.debug(*this);
    }

    void operator()() {}
    template<class Head, class... Args>
    void operator()(const Head& head, const Args&... args) {
        print(head);
        operator()(args...);
    }

    template<class T> Printer& operator<<(const T& a) {
        print(a);
        return *this;
    }

    Printer& operator<<(Printer& (*pf)(Printer&)) { return pf(*this); }
};

template<std::size_t buf_size, bool debug>
Printer<buf_size, debug>& endl(Printer<buf_size, debug>& pr) {
    pr.print_char('\n');
    pr.flush();
    return pr;
}
template<std::size_t buf_size, bool debug>
Printer<buf_size, debug>& flush(Printer<buf_size, debug>& pr) {
    pr.flush();
    return pr;
}

struct SetPrec {
    int n;
    template<class Pr> void print(Pr& pr) const { pr.set_decimal_precision(n); }
    template<class Pr> void debug(Pr& pr) const { pr.set_decimal_precision(n); }
};
SetPrec setprec(int n) { return SetPrec{n}; };

Printer<> print(1), eprint(2);

void prints() { print.print_char('\n'); }

template<class T> auto prints(const T& v) -> decltype(print << v, (void)0) {
    print << v;
    print.print_char('\n');
}

template<class Head, class... Tail>
auto prints(const Head& head, const Tail&... tail)
    -> decltype(print << head, (void)0) {
    print << head;
    print.print_char(' ');
    prints(tail...);
}

Printer<IO_BUFFER_SIZE, true> debug(1), edebug(2);

void debugs() { debug.print_char('\n'); }

template<class T> auto debugs(const T& v) -> decltype(debug << v, (void)0) {
    debug << v;
    debug.print_char('\n');
}

template<class Head, class... Tail>
auto debugs(const Head& head, const Tail&... tail)
    -> decltype(debug << head, (void)0) {
    debug << head;
    debug.print_char(' ');
    debugs(tail...);
}
#line 2 "template/bitop.hpp"

#line 6 "template/bitop.hpp"

namespace bitop {

#define KTH_BIT(b, k) (((b) >> (k)) & 1)
#define POW2(k) (1ull << (k))

inline ull next_combination(int n, ull x) {
    if (n == 0) return 1;
    ull a = x & -x;
    ull b = x + a;
    return (x & ~b) / a >> 1 | b;
}

#define rep_comb(i, n, k)                                                      \
    for (ull i = (1ull << (k)) - 1; i < (1ull << (n));                         \
         i = bitop::next_combination((n), i))

inline constexpr int msb(ull x) {
    int res = x ? 0 : -1;
    if (x & 0xFFFFFFFF00000000) x &= 0xFFFFFFFF00000000, res += 32;
    if (x & 0xFFFF0000FFFF0000) x &= 0xFFFF0000FFFF0000, res += 16;
    if (x & 0xFF00FF00FF00FF00) x &= 0xFF00FF00FF00FF00, res += 8;
    if (x & 0xF0F0F0F0F0F0F0F0) x &= 0xF0F0F0F0F0F0F0F0, res += 4;
    if (x & 0xCCCCCCCCCCCCCCCC) x &= 0xCCCCCCCCCCCCCCCC, res += 2;
    return res + ((x & 0xAAAAAAAAAAAAAAAA) ? 1 : 0);
}

inline constexpr int ceil_log2(ull x) { return x ? msb(x - 1) + 1 : 0; }

inline constexpr ull reverse(ull x) {
    x = ((x & 0xAAAAAAAAAAAAAAAA) >> 1) | ((x & 0x5555555555555555) << 1);
    x = ((x & 0xCCCCCCCCCCCCCCCC) >> 2) | ((x & 0x3333333333333333) << 2);
    x = ((x & 0xF0F0F0F0F0F0F0F0) >> 4) | ((x & 0x0F0F0F0F0F0F0F0F) << 4);
    x = ((x & 0xFF00FF00FF00FF00) >> 8) | ((x & 0x00FF00FF00FF00FF) << 8);
    x = ((x & 0xFFFF0000FFFF0000) >> 16) | ((x & 0x0000FFFF0000FFFF) << 16);
    return (x >> 32) | (x << 32);
}

inline constexpr ull reverse(ull x, int n) { return reverse(x) >> (64 - n); }

} // namespace bitop

inline constexpr int popcnt(ull x) noexcept {
#if __cplusplus >= 202002L
    return std::popcount(x);
#endif
    x = (x & 0x5555555555555555) + ((x >> 1) & 0x5555555555555555);
    x = (x & 0x3333333333333333) + ((x >> 2) & 0x3333333333333333);
    x = (x & 0x0f0f0f0f0f0f0f0f) + ((x >> 4) & 0x0f0f0f0f0f0f0f0f);
    x = (x & 0x00ff00ff00ff00ff) + ((x >> 8) & 0x00ff00ff00ff00ff);
    x = (x & 0x0000ffff0000ffff) + ((x >> 16) & 0x0000ffff0000ffff);
    return (x & 0x00000000ffffffff) + ((x >> 32) & 0x00000000ffffffff);
}
#line 2 "template/func.hpp"

#line 6 "template/func.hpp"

template<class T, class U, class Comp = std::less<>>
inline constexpr bool chmin(T& a, const U& b,
                            Comp cmp = Comp()) noexcept(noexcept(cmp(b, a))) {
    return cmp(b, a) ? a = b, true : false;
}
template<class T, class U, class Comp = std::less<>>
inline constexpr bool chmax(T& a, const U& b,
                            Comp cmp = Comp()) noexcept(noexcept(cmp(a, b))) {
    return cmp(a, b) ? a = b, true : false;
}

inline constexpr ll gcd(ll a, ll b) {
    if (a < 0) a = -a;
    if (b < 0) b = -b;
    while (b) {
        const ll c = a;
        a = b;
        b = c % b;
    }
    return a;
}
inline constexpr ll lcm(ll a, ll b) { return a / gcd(a, b) * b; }

inline constexpr bool is_prime(ll N) {
    if (N <= 1) return false;
    for (ll i = 2; i * i <= N; ++i) {
        if (N % i == 0) return false;
    }
    return true;
}
inline std::vector<ll> prime_factor(ll N) {
    std::vector<ll> res;
    for (ll i = 2; i * i <= N; ++i) {
        while (N % i == 0) {
            res.push_back(i);
            N /= i;
        }
    }
    if (N != 1) res.push_back(N);
    return res;
}

inline constexpr ll my_pow(ll a, ll b) {
    ll res = 1;
    while (b) {
        if (b & 1) res *= a;
        b >>= 1;
        a *= a;
    }
    return res;
}
inline constexpr ll mod_pow(ll a, ll b, ll mod) {
    assert(mod > 0);
    if (mod == 1) return 0;
    a %= mod;
    ll res = 1;
    while (b) {
        if (b & 1) (res *= a) %= mod;
        b >>= 1;
        (a *= a) %= mod;
    }
    return res;
}

inline PLL extGCD(ll a, ll b) {
    const ll n = a, m = b;
    ll x = 1, y = 0, u = 0, v = 1;
    ll t;
    while (b) {
        t = a / b;
        std::swap(a -= t * b, b);
        std::swap(x -= t * u, u);
        std::swap(y -= t * v, v);
    }
    if (x < 0) {
        x += m;
        y -= n;
    }
    return {x, y};
}
inline ll mod_inv(ll a, ll mod) {
    ll b = mod;
    ll x = 1, u = 0;
    ll t;
    while (b) {
        t = a / b;
        std::swap(a -= t * b, b);
        std::swap(x -= t * u, u);
    }
    if (x < 0) x += mod;
    assert(a == 1);
    return x;
}
#line 2 "template/util.hpp"

#line 6 "template/util.hpp"

template<class F> class RecLambda {
private:
    F f;

public:
    explicit constexpr RecLambda(F&& f_) : f(std::forward<F>(f_)) {}
    template<class... Args>
    constexpr auto operator()(Args&&... args)
        -> decltype(f(*this, std::forward<Args>(args)...)) {
        return f(*this, std::forward<Args>(args)...);
    }
};

template<class F> inline constexpr RecLambda<F> rec_lambda(F&& f) {
    return RecLambda<F>(std::forward<F>(f));
}


template<class Head, class... Tail> struct multi_dim_vector {
    using type = std::vector<typename multi_dim_vector<Tail...>::type>;
};
template<class T> struct multi_dim_vector<T> { using type = T; };

template<class T, class Arg>
constexpr std::vector<T> make_vec(int n, Arg&& arg) {
    return std::vector<T>(n, std::forward<Arg>(arg));
}
template<class T, class... Args>
constexpr typename multi_dim_vector<Args..., T>::type make_vec(int n,
                                                               Args&&... args) {
    return typename multi_dim_vector<Args..., T>::type(
        n, make_vec<T>(std::forward<Args>(args)...));
}


template<class T, class Comp = std::less<T>> class compressor {
private:
    std::vector<T> dat;
    Comp cmp;
    bool sorted = false;

public:
    compressor() : compressor(Comp()) {}
    compressor(const Comp& cmp) : cmp(cmp) {}
    compressor(const std::vector<T>& vec, bool f = false,
               const Comp& cmp = Comp())
        : dat(vec), cmp(cmp) {
        if (f) build();
    }
    compressor(std::vector<T>&& vec, bool f = false, const Comp& cmp = Comp())
        : dat(std::move(vec)), cmp(cmp) {
        if (f) build();
    }
    compressor(std::initializer_list<T> il, bool f = false,
               const Comp& cmp = Comp())
        : dat(all(il)), cmp(cmp) {
        if (f) build();
    }
    void reserve(int n) {
        assert(!sorted);
        dat.reserve(n);
    }
    void push_back(const T& v) {
        assert(!sorted);
        dat.push_back(v);
    }
    void push_back(T&& v) {
        assert(!sorted);
        dat.push_back(std::move(v));
    }
    template<class... Args> void emplace_back(Args&&... args) {
        assert(!sorted);
        dat.emplace_back(std::forward<Args>(args)...);
    }
    void push(const std::vector<T>& vec) {
        assert(!sorted);
        const int n = dat.size();
        dat.resize(n + vec.size());
        rep (i, vec.size()) dat[n + i] = vec[i];
    }
    int build() {
        assert(!sorted);
        sorted = true;
        std::sort(all(dat), cmp);
        dat.erase(std::unique(all(dat),
                              [&](const T& a, const T& b) -> bool {
                                  return !cmp(a, b) && !cmp(b, a);
                              }),
                  dat.end());
        return dat.size();
    }
    const T& operator[](int k) const& {
        assert(sorted);
        assert(0 <= k && k < (int)dat.size());
        return dat[k];
    }
    int get(const T& val) const {
        assert(sorted);
        auto itr = std::lower_bound(all(dat), val, cmp);
        assert(itr != dat.end() && !cmp(val, *itr));
        return itr - dat.begin();
    }
    int lower_bound(const T& val) const {
        assert(sorted);
        auto itr = std::lower_bound(all(dat), val, cmp);
        return itr - dat.begin();
    }
    int upper_bound(const T& val) const {
        assert(sorted);
        auto itr = std::upper_bound(all(dat), val, cmp);
        return itr - dat.begin();
    }
    bool contains(const T& val) const {
        assert(sorted);
        return std::binary_search(all(dat), val, cmp);
    }
    std::vector<int> pressed(const std::vector<T>& vec) const {
        assert(sorted);
        std::vector<int> res(vec.size());
        rep (i, vec.size()) res[i] = get(vec[i]);
        return res;
    }
    void press(std::vector<T>& vec) const {
        assert(sorted);
        for (auto&& i : vec) i = get(i);
    }
    int size() const {
        assert(sorted);
        return dat.size();
    }
};
#line 2 "other/monoid.hpp"

#line 4 "other/monoid.hpp"

namespace Monoid {

template<class M, class = void>
class has_value_type : public std::false_type {};
template<class M>
class has_value_type<M, decltype((void)std::declval<typename M::value_type>())>
    : public std::true_type {};

template<class M, class = void> class has_op : public std::false_type {};
template<class M>
class has_op<M, decltype((void)M::op)> : public std::true_type {};

template<class M, class = void> class has_id : public std::false_type {};
template<class M>
class has_id<M, decltype((void)M::id)> : public std::true_type {};

template<class M, class = void> class has_inv : public std::false_type {};
template<class M>
class has_inv<M, decltype((void)M::inv)> : public std::true_type {};

template<class M, class = void> class has_get_inv : public std::false_type {};
template<class M>
class has_get_inv<M, decltype((void)M::get_inv)> : public std::true_type {};

template<class M, class = void> class has_init : public std::false_type {};
template<class M>
class has_init<M, decltype((void)M::init(0, 0))> : public std::true_type {};

template<class A, class = void> class has_mul_op : public std::false_type {};
template<class A>
class has_mul_op<A, decltype((void)A::mul_op)> : public std::true_type {};

template<class T, class = void> class is_semigroup : public std::false_type {};
template<class T>
class is_semigroup<T, decltype(std::declval<typename T::value_type>(),
                               (void)T::op)> : public std::true_type {};

template<class T, class = void> class is_monoid : public std::false_type {};

template<class T>
class is_monoid<T, decltype(std::declval<typename T::value_type>(), (void)T::op,
                            (void)T::id)> : public std::true_type {};

template<class T, class = void> class is_group : public std::false_type {};

template<class T>
class is_group<T, decltype(std::declval<typename T::value_type>(), (void)T::op,
                           (void)T::id, (void)T::get_inv)>
    : public std::true_type {};

template<class T, class = void> class is_action : public std::false_type {};
template<class T>
class is_action<T, typename std::enable_if<is_monoid<typename T::M>::value &&
                                           is_semigroup<typename T::E>::value &&
                                           (has_op<T>::value ||
                                            has_mul_op<T>::value)>::type>
    : public std::true_type {};

template<class T, class = void>
class is_distributable_action : public std::false_type {};
template<class T>
class is_distributable_action<
    T,
    typename std::enable_if<is_action<T>::value && !has_mul_op<T>::value>::type>
    : public std::true_type {};

template<class T> struct Sum {
    using value_type = T;
    static constexpr T op(const T& a, const T& b) { return a + b; }
    static constexpr T id() { return T{0}; }
    static constexpr T inv(const T& a, const T& b) { return a - b; }
    static constexpr T get_inv(const T& a) { return -a; }
};

template<class T, int i = -1> struct Min {
    using value_type = T;
    static T max_value;
    static T op(const T& a, const T& b) { return a < b ? a : b; }
    static T id() { return max_value; }
};
template<class T> struct Min<T, -1> {
    using value_type = T;
    static constexpr T op(const T& a, const T& b) { return a < b ? a : b; }
    static constexpr T id() { return infinity<T>::value; }
};
template<class T> struct Min<T, -2> {
    using value_type = T;
    static constexpr T op(const T& a, const T& b) { return a < b ? a : b; }
    static constexpr T id() { return infinity<T>::max; }
};
template<class T, int id> T Min<T, id>::max_value;

template<class T, int i = -1> struct Max {
    using value_type = T;
    static T min_value;
    static T op(const T& a, const T& b) { return a > b ? a : b; }
    static T id() { return min_value; }
};
template<class T> struct Max<T, -1> {
    using value_type = T;
    static constexpr T op(const T& a, const T& b) { return a > b ? a : b; }
    static constexpr T id() { return infinity<T>::mvalue; }
};
template<class T> struct Max<T, -2> {
    using value_type = T;
    static constexpr T op(const T& a, const T& b) { return a > b ? a : b; }
    static constexpr T id() { return infinity<T>::min; }
};

template<class T> struct Assign {
    using value_type = T;
    static constexpr T op(const T&, const T& b) { return b; }
};


template<class T, int id = -1> struct AssignMin {
    using M = Min<T, id>;
    using E = Assign<T>;
    static constexpr T op(const T& a, const T&) { return a; }
};

template<class T, int id = -1> struct AssignMax {
    using M = Max<T, id>;
    using E = Assign<T>;
    static constexpr T op(const T& a, const T&) { return a; }
};

template<class T> struct AssignSum {
    using M = Sum<T>;
    using E = Assign<T>;
    static constexpr T mul_op(const T& a, int b, const T&) { return a * b; }
};

template<class T, int id = -1> struct AddMin {
    using M = Min<T, id>;
    using E = Sum<T>;
    static constexpr T op(const T& a, const T& b) { return b + a; }
};

template<class T, int id = -1> struct AddMax {
    using M = Max<T, id>;
    using E = Sum<T>;
    static constexpr T op(const T& a, const T& b) { return b + a; }
};

template<class T> struct AddSum {
    using M = Sum<T>;
    using E = Sum<T>;
    static constexpr T mul_op(const T& a, int b, const T& c) {
        return c + a * b;
    }
};

template<class T, int id = -1> struct ChminMin {
    using M = Min<T, id>;
    using E = Min<T>;
    static constexpr T op(const T& a, const T& b) { return std::min(b, a); }
};

template<class T, int id = -1> struct ChminMax {
    using M = Max<T, id>;
    using E = Min<T>;
    static constexpr T op(const T& a, const T& b) { return std::min(b, a); }
};

template<class T, int id = -1> struct ChmaxMin {
    using M = Min<T, id>;
    using E = Max<T>;
    static constexpr T op(const T& a, const T& b) { return std::max(b, a); }
};

template<class T, int id = -1> struct ChmaxMax {
    using M = Max<T, id>;
    using E = Max<T>;
    static constexpr T op(const T& a, const T& b) { return std::max(b, a); }
};


template<class M> struct ReverseMonoid {
    using value_type = typename M::value_type;
    static value_type op(const value_type& a, const value_type& b) {
        return M::op(b, a);
    }
    static value_type id() {
        static_assert(has_id<M>::value, "id is not defined");
        return M::id();
    }
    static value_type inv(const value_type& a, const value_type& b) {
        static_assert(has_inv<M>::value, "inv is not defined");
        return M::inv(b, a);
    }
    static value_type get_inv(const value_type& a) {
        static_assert(has_get_inv<M>::value, "get_inv is not defined");
        return M::get_inv(a);
    }
};

template<class E_> struct MakeAction {
    using M = E_;
    using E = E_;
    using T = typename E_::value_type;
    static T op(const T& a, const T& b) { return E_::op(b, a); }
};

} // namespace Monoid
#line 5 "data-struct/segment/DynamicLazySegmentTree.hpp"

template<class A> class DynamicLazySegmentTree {
private:
    using M = typename A::M;
    using E = typename A::E;
    using T = typename M::value_type;
    using U = typename E::value_type;
    struct node;
    using node_ptr = std::unique_ptr<node>;
    struct node {
        T val;
        U lazy;
        bool lazyflag;
        node_ptr l, r;
        node(const T& v) : val(v), lazyflag(false), l(nullptr), r(nullptr) {}
        node(const T& v, const U& x)
            : val(v), lazy(x), lazyflag(true), l(nullptr), r(nullptr) {}
    };
    node_ptr& get_l(const node_ptr& nd) const {
        if (nd->l == nullptr) nd->l = std::make_unique<node>(M::id());
        return nd->l;
    }
    node_ptr& get_r(const node_ptr& nd) const {
        if (nd->r == nullptr) nd->r = std::make_unique<node>(M::id());
        return nd->r;
    }

    template<bool AlwaysTrue = true,
             typename std::enable_if<!Monoid::has_mul_op<A>::value &&
                                     AlwaysTrue>::type* = nullptr>
    static inline T Aop(const U& a, const T& b, ll) {
        return A::op(a, b);
    }
    template<bool AlwaysTrue = true,
             typename std::enable_if<Monoid::has_mul_op<A>::value &&
                                     AlwaysTrue>::type* = nullptr>
    static inline T Aop(const U& a, const T& b, ll c) {
        return A::mul_op(a, c, b);
    }

    void all_apply(node_ptr& nd, int t, const U& x, ll d) {
        nd->val = Aop(x, nd->val, d);
        if (t != 0) {
            if (nd->lazyflag) {
                nd->lazy = E::op(nd->lazy, x);
            }
            else {
                nd->lazy = x;
                nd->lazyflag = true;
            }
        }
    }
    void eval(node_ptr& nd, ll a, ll b, int t) {
        if (nd->lazyflag) {
            ll m = (a + b) >> 1;
            all_apply(get_l(nd), t - 1, nd->lazy, m - a);
            all_apply(get_r(nd), t - 1, nd->lazy, b - m);
            nd->lazyflag = false;
        }
    }
    ll ori, h, n;
    std::vector<T> iv, iv2;
    node_ptr root;
    template<class Upd>
    void update(node_ptr& nd, ll a, ll b, int t, ll k, const Upd& upd) {
        if (a + 1 == b) {
            nd->val = upd(nd->val);
            return;
        }
        eval(nd, a, b, t);
        ll m = (a + b) >> 1;
        if (k < m) update(get_l(nd), a, m, t - 1, k, upd);
        else update(get_r(nd), m, b, t - 1, k, upd);
        nd->val =
            M::op(nd->l ? nd->l->val : M::id(), nd->r ? nd->r->val : M::id());
    }
    T prod(node_ptr& nd, ll a, ll b, int t, ll l, ll r) {
        if (nd == nullptr) return M::id();
        if (r <= a || b <= l) return M::id();
        if (l <= a && b <= r) return nd->val;
        eval(nd, a, b, t);
        ll m = (a + b) >> 1;
        return M::op(prod(nd->l, a, m, t - 1, l, r),
                     prod(nd->r, m, b, t - 1, l, r));
    }
    void apply(node_ptr& nd, ll a, ll b, int t, ll l, ll r, const U& x) {
        if (r <= a || b <= l) return;
        if (l <= a && b <= r) {
            all_apply(nd, t, x, b - a);
            return;
        }
        eval(nd, a, b, t);
        ll m = (a + b) >> 1;
        apply(get_l(nd), a, m, t - 1, l, r, x);
        apply(get_r(nd), m, b, t - 1, l, r, x);
        nd->val =
            M::op(nd->l ? nd->l->val : M::id(), nd->r ? nd->r->val : M::id());
    }
    template<class Cond>
    ll max_right(node_ptr& nd, ll a, ll b, int t, ll l, const Cond& cond,
                 T& sm) {
        if (b <= l || nd == nullptr) return n;
        if (l <= a && cond(M::op(sm, nd->val))) {
            sm = M::op(sm, nd->val);
            return n;
        }
        if (a + 1 == b) return a;
        eval(nd, a, b, t);
        ll m = (a + b) >> 1;
        ll res = max_right(nd->l, a, m, t - 1, l, cond, sm);
        if (res != n) return res;
        return max_right(nd->r, m, b, t - 1, l, cond, sm);
    }
    template<class Cond>
    ll min_left(node_ptr& nd, ll a, ll b, int t, ll r, const Cond& cond,
                T& sm) {
        if (r <= a || nd == nullptr) return 0;
        if (b <= r && cond(M::op(nd->val, sm))) {
            sm = M::op(nd->val, sm);
            return 0;
        }
        if (a + 1 == b) return b;
        eval(nd, a, b, t);
        ll m = (a + b) >> 1;
        ll res = min_left(nd->r, m, b, t - 1, r, cond, sm);
        if (res != 0) return res;
        return min_left(nd->l, a, m, t - 1, r, cond, sm);
    }
    void reset(node_ptr& nd, ll a, ll b, int t, ll l, ll r) {
        if (nd == nullptr) return;
        if (r <= a || b <= l) return;
        if (l <= a && b <= r) {
            if (nd == root) nd = std::make_unique<node>(M::id());
            else nd.reset();
            return;
        }
        ll m = (a + b) >> 1;
        reset(nd->l, a, m, t - 1, l, r);
        reset(nd->r, m, b, t - 1, l, r);
        nd->val =
            M::op(nd->l ? nd->l->val : M::id(), nd->r ? nd->r->val : M::id());
    }
    void merge(node_ptr& nd, ll a, ll b, int t, node_ptr& other) {
        if (nd == nullptr) {
            nd = std::move(other);
            return;
        }
        if (other == nullptr) return;
        if (nd->l == nullptr && nd->r == nullptr) {
            nd->val = M::op(nd->val, other->val);
            return;
        }
        eval(nd, a, b, t);
        eval(other, a, b, t);
        ll m = (a + b) >> 1;
        merge(nd->l, a, m, t - 1, other->l);
        merge(nd->r, m, b, t - 1, other->r);
        nd->val =
            M::op(nd->l ? nd->l->val : M::id(), nd->r ? nd->r->val : M::id());
    }
    void split(node_ptr& nd, ll a, ll b, int t, node_ptr& other, ll k) {
        if (nd == nullptr) return;
        if (a >= k) {
            other = std::move(nd);
            nd = nullptr;
            return;
        }
        if (b <= k) return;
        other = std::make_unique<node>(M::id());
        eval(nd, a, b, t);
        eval(other, a, b, t);
        ll m = (a + b) >> 1;
        split(nd->l, a, m, t - 1, other->l, k);
        split(nd->r, m, b, t - 1, other->r, k);
        if (nd->l == nullptr && nd->r == nullptr) nd = nullptr;
        else {
            nd->val = M::op(nd->l ? nd->l->val : M::id(),
                            nd->r ? nd->r->val : M::id());
        }
        if (other->l == nullptr && other->r == nullptr) other = nullptr;
        else {
            other->val = M::op(other->l ? other->l->val : M::id(),
                               other->r ? other->r->val : M::id());
        }
    }
    void init_copy(node_ptr& nd, const node_ptr& src) {
        if (src == nullptr) return;
        if (src->lazyflag) nd = std::make_unique<node>(src->val, src->lazy);
        else nd = std::make_unique<node>(src->val);
        init_copy(nd->l, src->l);
        init_copy(nd->r, src->r);
    }

public:
    DynamicLazySegmentTree() : DynamicLazySegmentTree(inf) {}
    DynamicLazySegmentTree(ll n_) { init(n_); }
    DynamicLazySegmentTree(ll n_, const T& v) { init(n_); }
    DynamicLazySegmentTree(const DynamicLazySegmentTree& other)
        : n(other.n), h(other.h), ori(other.ori), iv(other.iv), iv2(other.iv2),
          root(std::make_unique<node>(other.root->val)) {
        init_copy(root, other.root);
    }
    DynamicLazySegmentTree(DynamicLazySegmentTree&& other) = default;
    DynamicLazySegmentTree& operator=(const DynamicLazySegmentTree& other) {
        if (this == &other) return *this;
        return (*this) = DynamicLazySegmentTree(other);
    }
    DynamicLazySegmentTree& operator=(DynamicLazySegmentTree&& other) = default;
    void init(ll n_) {
        ori = n_;
        h = bitop::ceil_log2(ori);
        n = 1ull << h;
        root = std::make_unique<node>(M::id());
    }
    template<class Upd> void update(ll k, const Upd& upd) {
        assert(0 <= k && k < ori);
        update(root, 0, n, h, k, upd);
    }
    void set(ll k, T x) {
        update(k, [&](const T&) -> T { return x; });
    }
    void apply(ll k, const U& x) {
        update(k, [&](const T& a) -> T { return A::op(x, a); });
    }
    void apply(ll l, ll r, const U& x) {
        assert(0 <= l && l <= r && r <= ori);
        apply(root, 0, n, h, l, r, x);
    }
    void all_apply(const U& x) { apply(root, 0, n, h, 0, n, x); }
    T prod(ll l, ll r) {
        assert(0 <= l && l <= r && r <= ori);
        return prod(root, 0, n, h, l, r);
    }
    T all_prod() const { return root->val; }
    T get(ll k) { return prod(k, k + 1); }
    template<class Cond> ll max_right(ll l, const Cond& cond) {
        assert(0 <= l && l <= ori);
        if (l == n) return n;
        T sm = M::id();
        assert(cond(sm));
        return std::min(max_right(root, 0, n, h, l, cond, sm), ori);
    }
    template<class Cond> ll min_left(ll r, const Cond& cond) {
        assert(0 <= r && r <= ori);
        if (0 == r) return 0;
        T sm = M::id();
        assert(cond(sm));
        return min_left(root, 0, n, h, r, cond, sm);
    }
    void reset(ll l, ll r) { reset(root, 0, n, h, l, r); }
    void reset(ll k) { reset(root, 0, n, h, k, k + 1); }
    DynamicLazySegmentTree& merge(DynamicLazySegmentTree&& other) {
        assert(ori == other.ori);
        merge(root, 0, n, h, other.root);
        other.root = std::make_unique<node>(M::id());
        return *this;
    }
    friend DynamicLazySegmentTree merge(DynamicLazySegmentTree&& a,
                                        DynamicLazySegmentTree&& b) {
        return std::move(a.merge(std::move(b)));
    }
    friend std::pair<DynamicLazySegmentTree, DynamicLazySegmentTree>
    split(DynamicLazySegmentTree&& a, ll k) {
        assert(0 <= k && k <= a.ori);
        DynamicLazySegmentTree b(a.ori);
        a.split(a.root, 0, a.n, a.h, b.root, k);
        return {std::move(a), std::move(b)};
    }
};

/**
 * @brief DynamicLazySegmentTree(動的遅延セグメント木)
 * @docs docs/data-struct/segment/DynamicLazySegmentTree.md
 */

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