/**
 * 
 */

//#define NDEBUG

//#pragma GCC optimize("O3")
//#pragma GCC optimize("unroll-loops")

// atcoder
//#pragma GCC target("arch=ivybridge,tune=ivybridge")

// header {{{
#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/hash_policy.hpp>
#include <ext/pb_ds/priority_queue.hpp>
#include <ext/pb_ds/tag_and_trait.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace std;
namespace pbds = __gnu_pbds;

// C++20 polyfill {{{
struct IDENTITY {
    using is_transparent = void;
    template<typename T>
    constexpr T&& operator()(T&& x) const noexcept {
        return forward<T>(x);
    }
};
// }}}

#define CPP_STR(x) CPP_STR_I(x)
#define CPP_CAT(x,y) CPP_CAT_I(x,y)
#define CPP_STR_I(args...) #args
#define CPP_CAT_I(x,y) x ## y

#define SFINAE(pred...) std::enable_if_t<(pred), std::nullptr_t> = nullptr

#define ASSERT(expr...) assert((expr))

using i8   = int8_t;
using u8   = uint8_t;
using i16  = int16_t;
using u16  = uint16_t;
using i32  = int32_t;
using u32  = uint32_t;
using i64  = int64_t;
using u64  = uint64_t;
#ifdef __SIZEOF_INT128__
using i128 = __int128;
using u128 = unsigned __int128;
#endif

using f32  = float;
using f64  = double;
using f80  = __float80;
using f128 = __float128;

using complex32 = complex<f32>;
using complex64 = complex<f64>;
using complex80 = complex<f80>;
// }}}

template<typename T> constexpr T PROCON_INF();
template<> constexpr i64 PROCON_INF<i64>() { return 1'010'000'000'000'000'017LL; }
template<> constexpr f64 PROCON_INF<f64>() { return 1e100; }

constexpr i64 INF  = PROCON_INF<i64>();
constexpr f64 FINF = PROCON_INF<f64>();

constexpr i64 MOD = 1'000'000'007LL;

constexpr f64 EPS = 1e-12;

constexpr f64 PI = 3.14159265358979323846;

// util {{{
#define FOR(i, start, end) for(i64 i = (start), CPP_CAT(i,xxxx_end)=(end); i < CPP_CAT(i,xxxx_end); ++i)
#define REP(i, n) FOR(i, 0, n)

#define ALL(f,c,...) (([&](decltype((c)) cccc) { return (f)(std::begin(cccc), std::end(cccc), ## __VA_ARGS__); })(c))
#define SLICE(f,c,l,r,...) (([&](decltype((c)) cccc, decltype((l)) llll, decltype((r)) rrrr) {\
    auto iiii = llll <= rrrr ? std::begin(cccc)+llll : std::end(cccc);\
    auto jjjj = llll <= rrrr ? std::begin(cccc)+rrrr : std::end(cccc);\
    return (f)(iiii, jjjj, ## __VA_ARGS__);\
})(c,l,r))

#define GENERIC(f) ([](auto&&... args) -> decltype(auto) { return (f)(std::forward<decltype(args)>(args)...); })

// ビット演算 {{{
// 引数は [-INF,INF] のみ想定

i64 BIT_I(i64 i) {
    return 1LL << i;
}

i64 BIT_I_1(i64 i) {
    return BIT_I(i) - 1;
}

i64 BIT_GET(i64 x, i64 i) {
    return x & BIT_I(i);
}

bool BIT_TEST(i64 x, i64 i) {
    return BIT_GET(x,i) != 0;
}

i64 BIT_SET(i64 x, i64 i) {
    return x | BIT_I(i);
}

i64 BIT_CLEAR(i64 x, i64 i) {
    return x & ~BIT_I(i);
}

i64 BIT_FLIP(i64 x, i64 i) {
    return x ^ BIT_I(i);
}

i64 BIT_ASSIGN(i64 x, i64 i, bool b) {
    return b ? BIT_SET(x,i) : BIT_CLEAR(x,i);
}

i64 BIT_COUNT_LEADING_ZEROS(i64 x) {
    if(x == 0) return 64;
    return __builtin_clzll(x);
}

i64 BIT_COUNT_LEADING_ONES(i64 x) {
    return BIT_COUNT_LEADING_ZEROS(~x);
}

i64 BIT_COUNT_TRAILING_ZEROS(i64 x) {
    if(x == 0) return 64;
    return __builtin_ctzll(x);
}

i64 BIT_COUNT_TRAILING_ONES(i64 x) {
    return BIT_COUNT_TRAILING_ZEROS(~x);
}

// 末尾へ続く0を識別するマスクを返す (ex. 0b10100 -> 0b00011)
// x=0 なら -1 を返す
i64 BIT_MASK_TRAILING_ZEROS(i64 x) {
    return ~x & (x-1);
}

// 末尾へ続く1を識別するマスクを返す (ex. 0b10011 -> 0b00011)
// x=-1 なら -1 を返す
i64 BIT_MASK_TRAILING_ONES(i64 x) {
    return x & ~(x+1);
}

i64 BIT_COUNT_ONES(i64 x) {
    return __builtin_popcountll(x);
}

i64 BIT_COUNT_ZEROS(i64 x) {
    return 64 - BIT_COUNT_ONES(x);
}

// 先頭から続く冗長な符号ビットを数える (ex. 1 -> 62, -1 -> 63)
i64 BIT_COUNT_LEADING_REDUNDANT_SIGN_BITS(i64 x) {
    return __builtin_clrsbll(x);
}

// 1の個数が奇数なら1, 偶数なら0を返す
i64 BIT_PARITY(i64 x) {
    return __builtin_parityll(x);
}

// 最右の0を分離する (ex. 0b11001 -> 0b00010)
// x=-1 なら 0 を返す
i64 BIT_EXTRACT_FIRST_ZERO(i64 x) {
    return ~x & (x+1);
}

// 最右の1を分離する (ex. 0b10110 -> 0b00010)
// x=0 なら 0 を返す
i64 BIT_EXTRACT_FIRST_ONE(i64 x) {
    return x & (-x);
}

// 最右の0を1にする (ex. 0b11001 -> 0b11011)
i64 BIT_FLIP_FIRST_ZERO(i64 x) {
    return x | (x+1);
}

// 最右の1を0にする (ex. 0b10110 -> 0b10100)
i64 BIT_FLIP_FIRST_ONE(i64 x) {
    return x & (x-1);
}

// 最右の1の位置(1-based)を得る
// x=0 なら 0 を返す
i64 BIT_FIND_FIRST_ONE(i64 x) {
    return __builtin_ffsll(x);
}

// 最右の0の位置(1-based)を得る
// x=-1 なら 0 を返す
i64 BIT_FIND_FIRST_ZERO(i64 x) {
    return BIT_FIND_FIRST_ONE(~x);
}

// 最右の0をそれより右に伝播する (ex. 0b11011 -> 0b11000)
// x=-1 なら -1 を返す
i64 BIT_PROPAGATE_FIRST_ZERO(i64 x) {
    if(x == -1) return -1;
    return x & (x+1);
}

// 最右の1をそれより右に伝播する (ex. 0b10100 -> 0b10111)
// x=0 なら 0 を返す
i64 BIT_PROPAGATE_FIRST_ONE(i64 x) {
    if(x == 0) return 0;
    return x | (x-1);
}

// 最右の0および末尾へ続く1を識別するマスクを返す (ex. 0b11011 -> 0b00111)
// x=-1 なら 0 を返す
i64 BIT_MASKTO_FIRST_ZERO(i64 x) {
    if(x == -1) return 0;
    return x ^ (x+1);
}

// 最右の1および末尾へ続く0を識別するマスクを返す (ex. 0b10100 -> 0b00111)
// x=0 なら 0 を返す
i64 BIT_MASKTO_FIRST_ONE(i64 x) {
    if(x == 0) return 0;
    return x ^ (x-1);
}

// 最右の連続した0を1にする (ex. 0b101001 -> 0b101111)
// x=-1 なら -1 を返す
i64 BIT_FLIP_FIRST_ZEROS(i64 x) {
    return ((x&(x+1))-1) | x;
}

// 最右の連続した1を0にする (ex. 0b10110 -> 0b10000)
// x=0 なら 0 を返す
i64 BIT_FLIP_FIRST_ONES(i64 x) {
    return ((x|(x-1))+1) & x;
}

// X ⊆ {0,1,...,n-1}, |X| = k なる部分集合 X を昇順に列挙する
// comb(n,k) 個
//
// ex.
// ```
// i64 x = BIT_I_1(3);
// do {
//     // ...
// } while(BIT_NEXT_SET_SIZED(x, 10));
// ```
bool BIT_NEXT_SET_SIZED(i64& x, i64 n) {
    if(x == 0) return false;
    i64 t = BIT_PROPAGATE_FIRST_ONE(x) + 1;
    x = t | (BIT_MASK_TRAILING_ZEROS(t) >> (BIT_COUNT_TRAILING_ZEROS(x)+1));
    return x < BIT_I(n);
}

// 集合 Y の部分集合 X を昇順に列挙する
// 2^|Y| 個
//
// ex.
// ```
// i64 y = 0b10101;
// i64 x = 0;
// do {
//     // ...
// } while(BIT_NEXT_SUBSET(x, y));
// ```
bool BIT_NEXT_SUBSET(i64& x, i64 y) {
    if(x == y) return false;
    x = (x-y) & y;
    return true;
}

// 集合 Y の部分集合 X を降順に列挙する
// 2^|Y| 個
//
// ex.
// ```
// i64 y = 0b10101;
// i64 x = y;
// do {
//     // ...
// } while(BIT_PREV_SUBSET(x, y));
// ```
bool BIT_PREV_SUBSET(i64& x, i64 y) {
    if(x == 0) return false;
    x = (x-1) & y;
    return true;
}

// 集合 Y を包含する集合 X ⊆ {0,1,...,n-1} を昇順に列挙する
// 2^(n-|Y|) 個
//
// ex.
// ```
// i64 y = 0b00010101;
// i64 x = y;
// do {
//     // ...
// } while(BIT_NEXT_SUPERSET(x, 8, y));
// ```
bool BIT_NEXT_SUPERSET(i64& x, i64 n, i64 y) {
    x = (x+1) | y;
    return x < BIT_I(n);
}
// }}}

// BoolArray {{{
class BoolArray {
public:
    using value_type      = bool;
    using reference       = value_type&;
    using const_reference = const value_type&;
    using iterator        = value_type*;
    using const_iterator  = const value_type*;
    using difference_type = ptrdiff_t;
    using size_type       = size_t;

    using reverse_iterator       = std::reverse_iterator<iterator>;
    using const_reverse_iterator = std::reverse_iterator<const_iterator>;

    BoolArray() : BoolArray(0) {}
    explicit BoolArray(size_t n) : BoolArray(n,false) {}
    BoolArray(size_t n, bool value) : size_(n), data_(new bool[n]) {
        ALL(fill, *this, value);
    }

    BoolArray(initializer_list<bool> init) : size_(init.size()), data_(new bool[size_]) {
        ALL(copy, init, begin());
    }

    template<typename InputIt>
    BoolArray(InputIt first, InputIt last) {
        deque<bool> tmp(first, last);
        size_ = tmp.size();
        data_ = new bool[size_];
        ALL(copy, tmp, begin());
    }

    BoolArray(const BoolArray& other) : size_(other.size_), data_(new bool[size_]) {
        ALL(copy, other, begin());
    }

    BoolArray(BoolArray&& other) noexcept : size_(other.size_), data_(other.data_) {
        other.data_ = nullptr;
    }

    BoolArray& operator=(const BoolArray& other) {
        if(this == &other) return *this;
        if(!data_ || size_ < other.size_) {
            delete[] data_;
            data_ = new bool[other.size_];
        }
        size_ = other.size_;
        ALL(copy, other, begin());
        return *this;
    }

    BoolArray& operator=(BoolArray&& other) noexcept {
        if(this == &other) return *this;
        size_ = other.size_;
        data_ = other.data_;
        other.data_ = nullptr;
        return *this;
    }

    BoolArray& operator=(initializer_list<bool> init) {
        if(!data_ || size_ < init.size()) {
            delete[] data_;
            data_ = new bool[init.size()];
        }
        size_ = init.size();
        ALL(copy, init, begin());
        return *this;
    }

    void swap(BoolArray& other) noexcept {
        std::swap(size_, other.size_);
        std::swap(data_, other.data_);
    }

    ~BoolArray() {
        delete[] data_;
        data_ = nullptr;
    }

    bool      empty()    const noexcept { return size_ == 0; }
    size_type size()     const noexcept { return size_; }
    size_type max_size() const noexcept { return 1'010'000'000; }

    iterator       begin()        noexcept { return data_; }
    const_iterator begin()  const noexcept { return data_; }
    const_iterator cbegin() const noexcept { return data_; }

    iterator       end()        noexcept { return data_+size_; }
    const_iterator end()  const noexcept { return data_+size_; }
    const_iterator cend() const noexcept { return data_+size_; }

    reverse_iterator       rbegin()        noexcept { return reverse_iterator(end()); }
    const_reverse_iterator rbegin()  const noexcept { return const_reverse_iterator(end()); }
    const_reverse_iterator crbegin() const noexcept { return const_reverse_iterator(end()); }

    reverse_iterator       rend()        noexcept { return reverse_iterator(begin()); }
    const_reverse_iterator rend()  const noexcept { return const_reverse_iterator(begin()); }
    const_reverse_iterator crend() const noexcept { return const_reverse_iterator(begin()); }

    reference       operator[](size_type pos)       { return data_[pos]; }
    const_reference operator[](size_type pos) const { return data_[pos]; }

    bool*       data()       noexcept { return data_; }
    const bool* data() const noexcept { return data_; }

private:
    size_t size_;
    bool*  data_;
};

void swap(BoolArray& lhs, BoolArray& rhs) noexcept { lhs.swap(rhs); }

bool operator==(const BoolArray& lhs, const BoolArray& rhs) {
    return equal(begin(lhs), end(lhs), begin(rhs), end(rhs));
}
bool operator!=(const BoolArray& lhs, const BoolArray& rhs) { return !(lhs == rhs); }

bool operator<(const BoolArray& lhs, const BoolArray& rhs) {
    return lexicographical_compare(begin(lhs), end(lhs), begin(rhs), end(rhs));
}
bool operator> (const BoolArray& lhs, const BoolArray& rhs) { return rhs < lhs; }
bool operator<=(const BoolArray& lhs, const BoolArray& rhs) { return !(rhs < lhs); }
bool operator>=(const BoolArray& lhs, const BoolArray& rhs) { return !(lhs < rhs); }
// }}}

// 多次元 vector {{{
// 最内周が vector<bool> になるのを避けるための措置
template<typename T>
struct Array1Container {
    using type = vector<T>;
};
template<>
struct Array1Container<bool> {
    using type = BoolArray;
};

// イテレート用
template<typename T>
struct is_arrayn_container : false_type {};
template<typename T>
struct is_arrayn_container<vector<T>> : true_type {};
template<>
struct is_arrayn_container<BoolArray> : true_type {};

template<typename T>
auto arrayn_make(i64 n, T x) {
    using Cont = typename Array1Container<T>::type;
    return Cont(n, x);
}

template<typename T, typename... Args, SFINAE(sizeof...(Args) >= 2)>
auto arrayn_make(i64 n, Args... args) {
    auto inner = arrayn_make<T>(args...);
    return vector<decltype(inner)>(n, inner);
}

template<typename T, typename F, SFINAE(!is_arrayn_container<T>::value)>
void arrayn_foreach(T& e, F f) {
    f(e);
}

template<typename T, typename F, SFINAE(is_arrayn_container<T>::value)>
void arrayn_foreach(T& ary, F f) {
    for(auto& e : ary)
        arrayn_foreach(e, f);
}

template<typename T, typename U, SFINAE(is_arrayn_container<T>::value)>
void arrayn_fill(T& ary, const U& x) {
    arrayn_foreach(ary, [&x](auto& e) { e = x; });
}
// }}}

// 多次元生配列 {{{
template<typename T, typename F, SFINAE(rank<T>::value==0)>
void CARRAY_FOREACH(T& e, F f) {
    f(e);
}

template<typename Array, typename F, SFINAE(rank<Array>::value!=0)>
void CARRAY_FOREACH(Array& ary, F f) {
    for(auto& e : ary)
        CARRAY_FOREACH(e, f);
}

template<typename Array, typename U, SFINAE(rank<Array>::value!=0)>
void CARRAY_FILL(Array& ary, const U& v) {
    CARRAY_FOREACH(ary, [&v](auto& e) { e = v; });
}
// }}}

// メモ化ラッパー (8引数まで) {{{
template<i64 N1, typename F>
class Memoized1 {
    static_assert(N1 >= 1, "");
public:
    explicit Memoized1(F&& f) : f_(forward<F>(f)) {}
    decltype(auto) operator()(i64 x1) const {
        using R = decltype(f_(*this,x1));
        static bool done[N1] {};
        static R    memo[N1];
        if(!done[x1]) {
            memo[x1] = f_(*this,x1);
            done[x1] = true;
        }
        return memo[x1];
    }
private:
    const F f_;
};

template<i64 N1, i64 N2, typename F>
class Memoized2 {
    static_assert(N1 >= 1 && N2 >= 1, "");
public:
    explicit Memoized2(F&& f) : f_(forward<F>(f)) {}
    decltype(auto) operator()(i64 x1, i64 x2) const {
        using R = decltype(f_(*this,x1,x2));
        static bool done[N1][N2] {};
        static R    memo[N1][N2];
        if(!done[x1][x2]) {
            memo[x1][x2] = f_(*this,x1,x2);
            done[x1][x2] = true;
        }
        return memo[x1][x2];
    }
private:
    const F f_;
};

template<i64 N1, i64 N2, i64 N3, typename F>
class Memoized3 {
    static_assert(N1 >= 1 && N2 >= 1 && N3 >= 1, "");
public:
    explicit Memoized3(F&& f) : f_(forward<F>(f)) {}
    decltype(auto) operator()(i64 x1, i64 x2, i64 x3) const {
        using R = decltype(f_(*this,x1,x2,x3));
        static bool done[N1][N2][N3] {};
        static R    memo[N1][N2][N3];
        if(!done[x1][x2][x3]) {
            memo[x1][x2][x3] = f_(*this,x1,x2,x3);
            done[x1][x2][x3] = true;
        }
        return memo[x1][x2][x3];
    }
private:
    const F f_;
};

template<i64 N1, i64 N2, i64 N3, i64 N4, typename F>
class Memoized4 {
    static_assert(N1 >= 1 && N2 >= 1 && N3 >= 1 && N4 >= 1, "");
public:
    explicit Memoized4(F&& f) : f_(forward<F>(f)) {}
    decltype(auto) operator()(i64 x1, i64 x2, i64 x3, i64 x4) const {
        using R = decltype(f_(*this,x1,x2,x3,x4));
        static bool done[N1][N2][N3][N4] {};
        static R    memo[N1][N2][N3][N4];
        if(!done[x1][x2][x3][x4]) {
            memo[x1][x2][x3][x4] = f_(*this,x1,x2,x3,x4);
            done[x1][x2][x3][x4] = true;
        }
        return memo[x1][x2][x3][x4];
    }
private:
    const F f_;
};

template<i64 N1, i64 N2, i64 N3, i64 N4, i64 N5, typename F>
class Memoized5 {
    static_assert(N1 >= 1 && N2 >= 1 && N3 >= 1 && N4 >= 1 && N5 >= 1, "");
public:
    explicit Memoized5(F&& f) : f_(forward<F>(f)) {}
    decltype(auto) operator()(i64 x1, i64 x2, i64 x3, i64 x4, i64 x5) const {
        using R = decltype(f_(*this,x1,x2,x3,x4,x5));
        static bool done[N1][N2][N3][N4][N5] {};
        static R    memo[N1][N2][N3][N4][N5];
        if(!done[x1][x2][x3][x4][x5]) {
            memo[x1][x2][x3][x4][x5] = f_(*this,x1,x2,x3,x4,x5);
            done[x1][x2][x3][x4][x5] = true;
        }
        return memo[x1][x2][x3][x4][x5];
    }
private:
    const F f_;
};

template<i64 N1, i64 N2, i64 N3, i64 N4, i64 N5, i64 N6, typename F>
class Memoized6 {
    static_assert(N1 >= 1 && N2 >= 1 && N3 >= 1 && N4 >= 1 && N5 >= 1 && N6 >= 1, "");
public:
    explicit Memoized6(F&& f) : f_(forward<F>(f)) {}
    decltype(auto) operator()(i64 x1, i64 x2, i64 x3, i64 x4, i64 x5, i64 x6) const {
        using R = decltype(f_(*this,x1,x2,x3,x4,x5,x6));
        static bool done[N1][N2][N3][N4][N5][N6] {};
        static R    memo[N1][N2][N3][N4][N5][N6];
        if(!done[x1][x2][x3][x4][x5][x6]) {
            memo[x1][x2][x3][x4][x5][x6] = f_(*this,x1,x2,x3,x4,x5,x6);
            done[x1][x2][x3][x4][x5][x6] = true;
        }
        return memo[x1][x2][x3][x4][x5][x6];
    }
private:
    const F f_;
};

template<i64 N1, i64 N2, i64 N3, i64 N4, i64 N5, i64 N6, i64 N7, typename F>
class Memoized7 {
    static_assert(N1 >= 1 && N2 >= 1 && N3 >= 1 && N4 >= 1 && N5 >= 1 && N6 >= 1 && N7 >= 1, "");
public:
    explicit Memoized7(F&& f) : f_(forward<F>(f)) {}
    decltype(auto) operator()(i64 x1, i64 x2, i64 x3, i64 x4, i64 x5, i64 x6, i64 x7) const {
        using R = decltype(f_(*this,x1,x2,x3,x4,x5,x6,x7));
        static bool done[N1][N2][N3][N4][N5][N6][N7] {};
        static R    memo[N1][N2][N3][N4][N5][N6][N7];
        if(!done[x1][x2][x3][x4][x5][x6][x7]) {
            memo[x1][x2][x3][x4][x5][x6][x7] = f_(*this,x1,x2,x3,x4,x5,x6,x7);
            done[x1][x2][x3][x4][x5][x6][x7] = true;
        }
        return memo[x1][x2][x3][x4][x5][x6][x7];
    }
private:
    const F f_;
};

template<i64 N1, i64 N2, i64 N3, i64 N4, i64 N5, i64 N6, i64 N7, i64 N8, typename F>
class Memoized8 {
    static_assert(N1 >= 1 && N2 >= 1 && N3 >= 1 && N4 >= 1 && N5 >= 1 && N6 >= 1 && N7 >= 1 && N8 >= 1, "");
public:
    explicit Memoized8(F&& f) : f_(forward<F>(f)) {}
    decltype(auto) operator()(i64 x1, i64 x2, i64 x3, i64 x4, i64 x5, i64 x6, i64 x7, i64 x8) const {
        using R = decltype(f_(*this,x1,x2,x3,x4,x5,x6,x7,x8));
        static bool done[N1][N2][N3][N4][N5][N6][N7][N8] {};
        static R    memo[N1][N2][N3][N4][N5][N6][N7][N8];
        if(!done[x1][x2][x3][x4][x5][x6][x7][x8]) {
            memo[x1][x2][x3][x4][x5][x6][x7][x8] = f_(*this,x1,x2,x3,x4,x5,x6,x7,x8);
            done[x1][x2][x3][x4][x5][x6][x7][x8] = true;
        }
        return memo[x1][x2][x3][x4][x5][x6][x7][x8];
    }
private:
    const F f_;
};

template<i64 N1, typename F>
decltype(auto) MEMOIZE(F&& f) {
    return Memoized1<N1,F>(forward<F>(f));
}
template<i64 N1, i64 N2, typename F>
decltype(auto) MEMOIZE(F&& f) {
    return Memoized2<N1,N2,F>(forward<F>(f));
}
template<i64 N1, i64 N2, i64 N3, typename F>
decltype(auto) MEMOIZE(F&& f) {
    return Memoized3<N1,N2,N3,F>(forward<F>(f));
}
template<i64 N1, i64 N2, i64 N3, i64 N4, typename F>
decltype(auto) MEMOIZE(F&& f) {
    return Memoized4<N1,N2,N3,N4,F>(forward<F>(f));
}
template<i64 N1, i64 N2, i64 N3, i64 N4, i64 N5, typename F>
decltype(auto) MEMOIZE(F&& f) {
    return Memoized5<N1,N2,N3,N4,N5,F>(forward<F>(f));
}
template<i64 N1, i64 N2, i64 N3, i64 N4, i64 N5, i64 N6, typename F>
decltype(auto) MEMOIZE(F&& f) {
    return Memoized6<N1,N2,N3,N4,N5,N6,F>(forward<F>(f));
}
template<i64 N1, i64 N2, i64 N3, i64 N4, i64 N5, i64 N6, i64 N7, typename F>
decltype(auto) MEMOIZE(F&& f) {
    return Memoized7<N1,N2,N3,N4,N5,N6,N7,F>(forward<F>(f));
}
template<i64 N1, i64 N2, i64 N3, i64 N4, i64 N5, i64 N6, i64 N7, i64 N8, typename F>
decltype(auto) MEMOIZE(F&& f) {
    return Memoized8<N1,N2,N3,N4,N5,N6,N7,N8,F>(forward<F>(f));
}

// }}}

// lambda で再帰 {{{
template<typename F>
class FixPoint {
public:
    explicit constexpr FixPoint(F&& f) : f_(forward<F>(f)) {}

    template<typename... Args>
    constexpr decltype(auto) operator()(Args&&... args) const {
        return f_(*this, forward<Args>(args)...);
    }

private:
    const F f_;
};

template<typename F>
decltype(auto) FIX(F&& f) {
    return FixPoint<F>(forward<F>(f));
}
// }}}

// tuple {{{
template<typename... TS, SFINAE(sizeof...(TS) > 0)>
constexpr auto tuple_head(const tuple<TS...>& t) {
    return get<0>(t);
}

template<typename... TS, size_t i, size_t... is>
constexpr auto tuple_tail_helper(const tuple<TS...>& t, index_sequence<i,is...>) {
    return make_tuple(get<is>(t)...);
}

template<typename... TS, SFINAE(sizeof...(TS) == 1)>
constexpr auto tuple_tail(const tuple<TS...>&) {
    return make_tuple();
}

template<typename... TS, SFINAE(sizeof...(TS) > 1)>
constexpr auto tuple_tail(const tuple<TS...>& t) {
    return tuple_tail_helper(t, make_index_sequence<sizeof...(TS)>());
}

template<i64 I=0, typename F, typename... TS, SFINAE(sizeof...(TS) == I)>
void tuple_enumerate(tuple<TS...>&, F&&) {}

template<i64 I=0, typename F, typename... TS, SFINAE(sizeof...(TS) > I)>
void tuple_enumerate(tuple<TS...>& t, F&& f) {
    f(I, get<I>(t));
    tuple_enumerate<I+1>(t, f);
}

template<i64 I=0, typename F, typename... TS, SFINAE(sizeof...(TS) == I)>
void tuple_enumerate(const tuple<TS...>&, F&&) {}

template<i64 I=0, typename F, typename... TS, SFINAE(sizeof...(TS) > I)>
void tuple_enumerate(const tuple<TS...>& t, F&& f) {
    f(I, get<I>(t));
    tuple_enumerate<I+1>(t, f);
}
// }}}

// FST/SND {{{
template<typename T1, typename T2>
T1& FST(pair<T1,T2>& p) {
    return p.first;
}

template<typename T1, typename T2>
const T1& FST(const pair<T1,T2>& p) {
    return p.first;
}

template<typename T1, typename T2>
T2& SND(pair<T1,T2>& p) {
    return p.second;
}

template<typename T1, typename T2>
const T2& SND(const pair<T1,T2>& p) {
    return p.second;
}

template<typename... TS, SFINAE(sizeof...(TS) >= 1)>
auto& FST(tuple<TS...>& t) {
    return get<0>(t);
}

template<typename... TS, SFINAE(sizeof...(TS) >= 1)>
const auto& FST(const tuple<TS...>& t) {
    return get<0>(t);
}

template<typename... TS, SFINAE(sizeof...(TS) >= 2)>
auto& SND(tuple<TS...>& t) {
    return get<1>(t);
}

template<typename... TS, SFINAE(sizeof...(TS) >= 2)>
const auto& SND(const tuple<TS...>& t) {
    return get<1>(t);
}
// }}}

template<typename T1, typename T2, typename Comp=less<>,
         SFINAE(
             is_integral<T1>::value &&
             is_integral<T2>::value &&
             is_signed<T1>::value != is_unsigned<T2>::value
         )>
common_type_t<T1,T2> MAX(T1 x, T2 y, Comp comp={}) {
    return max<common_type_t<T1,T2>>(x, y, comp);
}

template<typename T1, typename T2, typename Comp=less<>,
         SFINAE(
             is_floating_point<T1>::value &&
             is_floating_point<T2>::value
         )>
common_type_t<T1,T2> MAX(T1 x, T2 y, Comp comp={}) {
    return max<common_type_t<T1,T2>>(x, y, comp);
}

template<typename T, typename Comp=less<>>
const T& MAX(const T& x, const T& y, Comp comp={}) {
    return max(x, y, comp);
}

template<typename T, typename Comp=less<>>
T MAX(initializer_list<T> ilist, Comp comp={}) {
    return max(ilist, comp);
}

template<typename T1, typename T2, typename Comp=less<>,
         SFINAE(
             is_integral<T1>::value &&
             is_integral<T2>::value &&
             is_signed<T1>::value != is_unsigned<T2>::value
         )>
common_type_t<T1,T2> MIN(T1 x, T2 y, Comp comp={}) {
    return min<common_type_t<T1,T2>>(x, y, comp);
}

template<typename T1, typename T2, typename Comp=less<>,
         SFINAE(
             is_floating_point<T1>::value &&
             is_floating_point<T2>::value
         )>
common_type_t<T1,T2> MIN(T1 x, T2 y, Comp comp={}) {
    return min<common_type_t<T1,T2>>(x, y, comp);
}

template<typename T, typename Comp=less<>>
const T& MIN(const T& x, const T& y, Comp comp={}) {
    return min(x, y, comp);
}

template<typename T, typename Comp=less<>>
T MIN(initializer_list<T> ilist, Comp comp={}) {
    return min(ilist, comp);
}

template<typename T1, typename T2, typename T3, typename Comp=less<>, SFINAE(
    is_integral<T1>::value &&
    is_integral<T2>::value &&
    is_integral<T3>::value &&
    is_signed<T1>::value != is_unsigned<T2>::value &&
    is_signed<T2>::value != is_unsigned<T3>::value
)>
common_type_t<T1,T2,T3> CLAMP(T1 x, T2 xmin, T3 xmax, Comp comp={}) {
    ASSERT(!comp(xmax, xmin));
    if(comp(x, xmin)) return xmin;
    if(comp(xmax, x)) return xmax;
    return x;
}

template<typename T1, typename T2, typename T3, typename Comp=less<>, SFINAE(
    is_floating_point<T1>::value &&
    is_floating_point<T2>::value &&
    is_floating_point<T3>::value
)>
common_type_t<T1,T2,T3> CLAMP(T1 x, T2 xmin, T3 xmax, Comp comp={}) {
    ASSERT(!comp(xmax, xmin));
    if(comp(x, xmin)) return xmin;
    if(comp(xmax, x)) return xmax;
    return x;
}

template<typename T, typename Comp=less<>>
const T& CLAMP(const T& x, const T& xmin, const T& xmax, Comp comp={}) {
    ASSERT(!comp(xmax, xmin));
    if(comp(x, xmin)) return xmin;
    if(comp(xmax, x)) return xmax;
    return x;
}

template<typename T>
T ABS(T x) {
    static_assert(is_signed<T>::value, "ABS(): argument must be signed");
    return x < 0 ? -x : x;
}

f64 ROUND(f64 x) {
    return round(x);
}

i64 IROUND(f64 x) {
    return llround(x);
}

template<typename C>
i64 SIZE(const C& c) { return static_cast<i64>(c.size()); }

template<typename T, size_t N>
i64 SIZE(const T (&)[N]) { return static_cast<i64>(N); }

bool is_odd (i64 x) { return x % 2 != 0; }
bool is_even(i64 x) { return x % 2 == 0; }

template<typename T> i64 cmp(T x, T y) { return (y<x) - (x<y); }
template<typename T> i64 sgn(T x) { return cmp(x, T(0)); }

// 事前条件: a >= 0, b >= 0
i64 gcd_impl(i64 a, i64 b) {
    if(b == 0) return a;
    return gcd_impl(b, a%b);
}

// GCD(0,0) = 0
i64 GCD(i64 a, i64 b) {
    return gcd_impl(ABS(a), ABS(b));
}

// LCM(0,x) は未定義
i64 LCM(i64 a, i64 b) {
    ASSERT(a != 0 && b != 0);
    a = ABS(a);
    b = ABS(b);
    return a / gcd_impl(a,b) * b;
}

// lo:OK, hi:NG
template<typename Pred>
i64 bisect_integer(i64 lo, i64 hi, Pred pred) {
    ASSERT(lo < hi);

    while(lo+1 < hi) {
        i64 mid = (lo+hi) / 2;
        if(pred(mid))
            lo = mid;
        else
            hi = mid;
    }
    return lo;
}

template<typename Pred>
f64 bisect_real(f64 lo, f64 hi, Pred pred, i64 iter=100) {
    ASSERT(lo < hi);

    REP(_, iter) {
        f64 mid = (lo+hi) / 2;
        if(pred(mid))
            lo = mid;
        else
            hi = mid;
    }
    return lo;
}

i64 ipow(i64 x, i64 e) {
    ASSERT(e >= 0);
    i64 res = 1;
    REP(_, e) {
        res *= x;
    }
    return res;
}

i64 sqrt_floor(i64 x) {
    ASSERT(x >= 0);

    i64 lo = 0;
    i64 hi = MIN(x/2+2, 3037000500LL);
    return bisect_integer(lo, hi, [x](i64 r) { return r*r <= x; });
}

i64 sqrt_ceil(i64 x) {
    i64 r = sqrt_floor(x);
    return r*r == x ? r : r+1;
}

// 0 <= log2_ceil(x) <= 63
i64 log2_ceil(i64 x) {
    ASSERT(x > 0);
    return 64 - BIT_COUNT_LEADING_ZEROS(x-1);
}

// 0 <= log2_floor(x) <= 62
i64 log2_floor(i64 x) {
    ASSERT(x > 0);
    return 63 - BIT_COUNT_LEADING_ZEROS(x);
}

// 0 <= log10_ceil(x) <= 19
i64 log10_ceil(i64 x) {
    ASSERT(x > 0);
    static constexpr i8 TABLE1[64] {
        -1, 19, 19, 19, 19, 18, 18, 18,
        17, 17, 17, 16, 16, 16, 16, 15,
        15, 15, 14, 14, 14, 13, 13, 13,
        13, 12, 12, 12, 11, 11, 11, 10,
        10, 10, 10,  9,  9,  9,  8,  8,
         8,  7,  7,  7,  7,  6,  6,  6,
         5,  5,  5,  4,  4,  4,  4,  3,
         3,  3,  2,  2,  2,  1,  1,  0,
    };
    static constexpr i64 TABLE2[20] {
        0LL,
        1LL,
        10LL,
        100LL,
        1000LL,
        10000LL,
        100000LL,
        1000000LL,
        10000000LL,
        100000000LL,
        1000000000LL,
        10000000000LL,
        100000000000LL,
        1000000000000LL,
        10000000000000LL,
        100000000000000LL,
        1000000000000000LL,
        10000000000000000LL,
        100000000000000000LL,
        1000000000000000000LL,
    };
    i64 res = TABLE1[BIT_COUNT_LEADING_ZEROS(x)];
    if(x <= TABLE2[res]) --res;
    return res;
}

// 0 <= log10_floor(x) <= 18
i64 log10_floor(i64 x) {
    ASSERT(x > 0);
    static constexpr i8 TABLE1[64] {
        -1, 18, 18, 18, 18, 17, 17, 17,
        16, 16, 16, 15, 15, 15, 15, 14,
        14, 14, 13, 13, 13, 12, 12, 12,
        12, 11, 11, 11, 10, 10, 10,  9,
         9,  9,  9,  8,  8,  8,  7,  7,
         7,  6,  6,  6,  6,  5,  5,  5,
         4,  4,  4,  3,  3,  3,  3,  2,
         2,  2,  1,  1,  1,  0,  0,  0,
    };
    static constexpr i64 TABLE2[19] {
        1LL,
        10LL,
        100LL,
        1000LL,
        10000LL,
        100000LL,
        1000000LL,
        10000000LL,
        100000000LL,
        1000000000LL,
        10000000000LL,
        100000000000LL,
        1000000000000LL,
        10000000000000LL,
        100000000000000LL,
        1000000000000000LL,
        10000000000000000LL,
        100000000000000000LL,
        1000000000000000000LL,
    };
    i64 res = TABLE1[BIT_COUNT_LEADING_ZEROS(x)];
    if(x < TABLE2[res]) --res;
    return res;
}

// 2^n - 1 の形かどうか
bool is_mersenne(i64 x) {
    ASSERT(x >= 0);
    return (x&(x+1)) == 0;
}

bool is_pow2(i64 x) {
    ASSERT(x > 0);
    return (x&(x-1)) == 0;
}

// x > 0
i64 pow2_ceil(i64 x) {
    return BIT_I(log2_ceil(x));
}

// x > 0
i64 pow2_floor(i64 x) {
    return BIT_I(log2_floor(x));
}

// Haskell の divMod と同じ
pair<i64,i64> divmod(i64 a, i64 b) {
    i64 q = a / b;
    i64 r = a % b;
    if((b>0 && r<0) || (b<0 && r>0)) {
        --q;
        r += b;
    }
    return {q,r};
}

i64 div_ceil(i64 a, i64 b) {
    i64 q = a / b;
    i64 r = a % b;
    if((b>0 && r>0) || (b<0 && r<0))
        ++q;
    return q;
}

i64 div_floor(i64 a, i64 b) {
    return divmod(a,b).first;
}

i64 modulo(i64 a, i64 b) {
    return divmod(a,b).second;
}

// x を align の倍数に切り上げる
i64 align_ceil(i64 x, i64 align) {
    ASSERT(align > 0);
    return div_ceil(x,align) * align;
}

// x を align の倍数に切り下げる
i64 align_floor(i64 x, i64 align) {
    ASSERT(align > 0);
    return div_floor(x,align) * align;
}

bool feq(f64 x, f64 y, f64 eps=EPS) {
    return fabs(x-y) < eps;
}

template<typename T, typename U, typename Comp=less<>>
bool chmax(T& xmax, const U& x, Comp comp={}) {
    if(comp(xmax, x)) {
        xmax = x;
        return true;
    }
    return false;
}

template<typename T, typename U, typename Comp=less<>>
bool chmin(T& xmin, const U& x, Comp comp={}) {
    if(comp(x, xmin)) {
        xmin = x;
        return true;
    }
    return false;
}

template<typename Pred>
i64 arg_find(i64 lo, i64 hi, Pred pred) {
    ASSERT(lo < hi);

    FOR(x, lo, hi) {
        if(pred(x)) return x;
    }
    return INF;
}

template<typename F>
i64 arg_max(i64 lo, i64 hi, F f) {
    ASSERT(lo < hi);

    i64 res = lo;
    auto ymax = f(lo);
    FOR(x, lo+1, hi) {
        if(chmax(ymax, f(x)))
            res = x;
    }
    return res;
}

template<typename F>
i64 arg_min(i64 lo, i64 hi, F f) {
    ASSERT(lo < hi);

    i64 res = lo;
    auto ymin = f(lo);
    FOR(x, lo+1, hi) {
        if(chmin(ymin, f(x)))
            res = x;
    }
    return res;
}

template<typename Pred>
i64 arg_find_r(i64 lo, i64 hi, Pred pred) {
    i64 x = arg_find(-hi+1, lo+1, [pred](i64 xx) { return pred(-xx); });
    return x == INF ? INF : -x;
}

template<typename F>
i64 arg_max_r(i64 lo, i64 hi, F f) {
    return -arg_max(-hi+1, lo+1, [f](i64 x) { return f(-x); });
}

template<typename F>
i64 arg_min_r(i64 lo, i64 hi, F f) {
    return -arg_min(-hi+1, lo+1, [f](i64 x) { return f(-x); });
}

template<typename ForwardIt, typename T, typename Comp=less<>>
ForwardIt bsearch_find(ForwardIt first, ForwardIt last, const T& x, Comp comp={}) {
    auto it = lower_bound(first, last, x, comp);
    if(it == last || comp(x,*it)) return last;
    return it;
}

// x 未満の最後の要素
template<typename BidiIt, typename T, typename Comp=less<>>
BidiIt bsearch_lt(BidiIt first, BidiIt last, const T& x, Comp comp={}) {
    auto it = lower_bound(first, last, x, comp);
    if(it == first) return last;
    return prev(it);
}

// x 以下の最後の要素
template<typename BidiIt, typename T, typename Comp=less<>>
BidiIt bsearch_le(BidiIt first, BidiIt last, const T& x, Comp comp={}) {
    auto it = upper_bound(first, last, x, comp);
    if(it == first) return last;
    return prev(it);
}

// x より大きい最初の要素
template<typename BidiIt, typename T, typename Comp=less<>>
BidiIt bsearch_gt(BidiIt first, BidiIt last, const T& x, Comp comp={}) {
    return upper_bound(first, last, x, comp);
}

// x 以上の最初の要素
template<typename BidiIt, typename T, typename Comp=less<>>
BidiIt bsearch_ge(BidiIt first, BidiIt last, const T& x, Comp comp={}) {
    return lower_bound(first, last, x, comp);
}

template<typename InputIt, typename BinaryOp>
auto FOLD(InputIt first, InputIt last,
          typename iterator_traits<InputIt>::value_type init,
          BinaryOp op)
{
    for(; first != last; ++first)
        init = op(move(init), *first);
    return init;
}

template<typename InputIt, typename BinaryOp>
auto FOLD1(InputIt first, InputIt last, BinaryOp op) {
    auto init = *first++;
    return FOLD(first, last, init, op);
}

template<typename InputIt>
auto SUM(InputIt first, InputIt last) {
    using T = typename iterator_traits<InputIt>::value_type;
    return accumulate(first, last, T());
}

template<typename ForwardIt, typename UnaryOperation>
ForwardIt transform_self(ForwardIt first, ForwardIt last, UnaryOperation op) {
    return transform(first, last, first, op);
}

template<typename C>
void UNIQ(C& c) {
    c.erase(ALL(unique,c), end(c));
}

template<typename BinaryFunc>
auto FLIP(BinaryFunc f) {
    return [f](const auto& x, const auto& y) {
        return f(y,x);
    };
}

template<typename BinaryFunc, typename UnaryFunc>
auto ON(BinaryFunc bf, UnaryFunc uf) {
    return [bf,uf](const auto& x, const auto& y) {
        return bf(uf(x), uf(y));
    };
}

template<typename F>
auto LT_ON(F f) { return ON(less<>(), f); }

template<typename F>
auto GT_ON(F f) { return ON(greater<>(), f); }

template<typename F>
auto EQ_ON(F f) { return ON(equal_to<>(), f); }

template<typename F>
auto NE_ON(F f) { return ON(not_equal_to<>(), f); }

template<typename Comp=less<>>
auto EQUIV(Comp comp={}) {
    return [comp](const auto& lhs, const auto& rhs) {
        return !comp(lhs,rhs) && !comp(rhs,lhs);
    };
}

char digit_chr(i64 n) {
    return static_cast<char>('0' + n);
}

i64 digit_ord(char c) {
    return c - '0';
}

char lower_chr(i64 n) {
    return static_cast<char>('a' + n);
}

i64 lower_ord(char c) {
    return c - 'a';
}

char upper_chr(i64 n) {
    return static_cast<char>('A' + n);
}

i64 upper_ord(char c) {
    return c - 'A';
}

// 出力は operator<< を直接使わず、このテンプレート経由で行う
// 提出用出力とデバッグ用出力を分けるため
template<typename T, typename Enable=void>
struct Formatter {
    static ostream& write_str(ostream& out, const T& x)  { return out << x; }
    static ostream& write_repr(ostream& out, const T& x) { return out << x; }
};

template<typename T>
ostream& WRITE_STR(ostream& out, const T& x) {
    return Formatter<T>::write_str(out, x);
}

template<typename T>
ostream& WRITE_REPR(ostream& out, const T& x) {
    return Formatter<T>::write_repr(out, x);
}

template<typename InputIt>
ostream& WRITE_JOIN_STR(ostream& out, InputIt first, InputIt last, const string& sep) {
    while(first != last) {
        WRITE_STR(out, *first++);
        if(first != last)
            out << sep;
    }
    return out;
}

template<typename InputIt>
ostream& WRITE_JOIN_REPR(ostream& out, InputIt first, InputIt last, const string& sep) {
    while(first != last) {
        WRITE_REPR(out, *first++);
        if(first != last)
            out << sep;
    }
    return out;
}

template<typename InputIt>
ostream& WRITE_RANGE_STR(ostream& out, InputIt first, InputIt last) {
    return WRITE_JOIN_STR(out, first, last, " ");
}

template<typename InputIt>
ostream& WRITE_RANGE_REPR(ostream& out, InputIt first, InputIt last) {
    out << "[";
    WRITE_JOIN_REPR(out, first, last, ", ");
    out << "]";
    return out;
}

template<typename T>
string TO_STR(const T& x) {
    ostringstream out;
    WRITE_STR(out, x);
    return out.str();
}

template<typename T>
string TO_REPR(const T& x) {
    ostringstream out;
    WRITE_REPR(out, x);
    return out.str();
}

template<typename InputIt>
string RANGE_TO_STR(InputIt first, InputIt last) {
    ostringstream out;
    WRITE_RANGE_STR(out, first, last);
    return out.str();
}

template<typename InputIt>
string RANGE_TO_REPR(InputIt first, InputIt last) {
    ostringstream out;
    WRITE_RANGE_REPR(out, first, last);
    return out.str();
}

template<typename InputIt>
string JOIN(InputIt first, InputIt last, const string& sep) {
    ostringstream out;
    WRITE_JOIN_STR(out, first, last, sep);
    return out.str();
}

template<>
struct Formatter<i64> {
    static ostream& write_str(ostream& out, i64 x) {
        return out << x;
    }
    static ostream& write_repr(ostream& out, i64 x) {
        if(x == INF) return out << "INF";
        if(x == -INF) return out << "-INF";
        return out << x;
    }
};

template<>
struct Formatter<f64> {
    static ostream& write_str(ostream& out, f64 x) {
        return out << x;
    }
    static ostream& write_repr(ostream& out, f64 x) {
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
        if(x == FINF) return out << "FINF";
        if(x == -FINF) return out << "-FINF";
#pragma GCC diagnostic pop
        return out << x;
    }
};

template<typename Enum>
struct Formatter<Enum, enable_if_t<is_enum<Enum>::value>> {
    static ostream& write_str(ostream& out, Enum x) {
        return WRITE_STR(out, static_cast<underlying_type_t<Enum>>(x));
    }
    static ostream& write_repr(ostream& out, Enum x) {
        return WRITE_REPR(out, static_cast<underlying_type_t<Enum>>(x));
    }
};

template<typename T>
struct Formatter<vector<T>> {
    static ostream& write_str(ostream& out, const vector<T>& v) {
        return WRITE_RANGE_STR(out, begin(v), end(v));
    }
    static ostream& write_repr(ostream& out, const vector<T>& v) {
        out << "vector";
        return WRITE_RANGE_REPR(out, begin(v), end(v));
    }
};

template<>
struct Formatter<vector<string>> {
    static ostream& write_str(ostream& out, const vector<string>& v) {
        for(const auto& row : v) {
            WRITE_STR(out, row);
            out << "\n";
        }
        return out;
    }
    static ostream& write_repr(ostream& out, const vector<string>& v) {
        out << "\n";
        for(const auto& row : v) {
            WRITE_STR(out, row);
            out << "\n";
        }
        return out;
    }
};

template<>
struct Formatter<BoolArray> {
    static ostream& write_str(ostream& out, const BoolArray& a) {
        return WRITE_RANGE_STR(out, begin(a), end(a));
    }
    static ostream& write_repr(ostream& out, const BoolArray& a) {
        out << "BoolArray";
        return WRITE_RANGE_REPR(out, begin(a), end(a));
    }
};

template<typename T1, typename T2>
struct Formatter<pair<T1,T2>> {
    static ostream& write_str(ostream& out, const pair<T1,T2>& p) {
        WRITE_STR(out, p.first);
        out << ' ';
        WRITE_STR(out, p.second);
        return out;
    }
    static ostream& write_repr(ostream& out, const pair<T1,T2>& p) {
        out << "(";
        WRITE_REPR(out, p.first);
        out << ",";
        WRITE_REPR(out, p.second);
        out << ")";
        return out;
    }
};

template<typename... TS>
struct Formatter<tuple<TS...>> {
    static ostream& write_str(ostream& out, const tuple<TS...>& t) {
        tuple_enumerate(t, [&out](i64 i, const auto& e) {
            if(i != 0) out << ' ';
            WRITE_STR(out, e);
        });
        return out;
    }
    static ostream& write_repr(ostream& out, const tuple<TS...>& t) {
        out << "(";
        tuple_enumerate(t, [&out](i64 i, const auto& e) {
            if(i != 0) out << ",";
            WRITE_REPR(out, e);
        });
        out << ")";
        return out;
    }
};

template<typename T, typename Enable=void>
struct Scanner {
    static_assert(!is_same<T,bool>::value, "Scanner<bool> is not supported");
    static T read(istream& in) {
        T res;
        in >> res;
        return res;
    }
};

template<typename T>
struct Scanner<T, enable_if_t<is_integral<T>::value && !is_same<T,bool>::value>> {
    static T read(istream& in) {
        T res;
        in >> res;
        return res;
    }
    static T read1(istream& in) {
        return read(in) - 1;
    }
};

template<typename T>
T READ(istream& in) {
    return Scanner<T>::read(in);
}

template<typename T>
T READ1(istream& in) {
    return Scanner<T>::read1(in);
}

template<typename T>
T FROM_STR(const string& s) {
    istringstream in(s);
    return READ<T>(in);
}

template<typename T=i64>
T RD() {
    return READ<T>(cin);
}

template<typename T=i64>
T RD1() {
    return READ1<T>(cin);
}

template<typename T=i64>
auto RD_ARRAY(i64 n) {
    vector<T> res;
    res.reserve(n);
    REP(_, n) {
        res.emplace_back(RD<T>());
    }
    return res;
}

template<typename T=i64>
auto RD1_ARRAY(i64 n) {
    vector<T> res;
    res.reserve(n);
    REP(_, n) {
        res.emplace_back(RD1<T>());
    }
    return res;
}

template<typename T=i64>
auto RD_ARRAY2(i64 h, i64 w) {
    vector<vector<T>> res(h);
    for(auto& row : res) {
        row.reserve(w);
        REP(_, w) {
            row.emplace_back(RD<T>());
        }
    }
    return res;
}

template<typename T=i64>
auto RD1_ARRAY2(i64 h, i64 w) {
    vector<vector<T>> res(h);
    for(auto& row : res) {
        row.reserve(w);
        REP(_, w) {
            row.emplace_back(RD1<T>());
        }
    }
    return res;
}

template<typename T1, typename T2>
struct Scanner<pair<T1,T2>> {
    static pair<T1,T2> read(istream& in) {
        T1 x = READ<T1>(in);
        T2 y = READ<T2>(in);
        return {x,y};
    }
};

template<typename... TS>
struct Scanner<tuple<TS...>> {
    template<i64 I, SFINAE(sizeof...(TS) == I)>
    static auto read_impl(istream&) {
        return make_tuple();
    }
    template<i64 I, SFINAE(sizeof...(TS) > I)>
    static auto read_impl(istream& in) {
        using T = tuple_element_t<I,tuple<TS...>>;
        auto head = make_tuple(READ<T>(in));
        return tuple_cat(head, read_impl<I+1>(in));
    }

    static tuple<TS...> read(istream& in) {
        return read_impl<0>(in);
    }
};

void PRINT() {}

template<typename T, typename... TS>
void PRINT(const T& x, const TS& ...args) {
    WRITE_STR(cout, x);
    if(sizeof...(args)) {
        cout << ' ';
        PRINT(args...);
    }
}

template<typename... TS>
void PRINTLN(const TS& ...args) {
    PRINT(args...);
    cout << '\n';
}

[[noreturn]] void EXIT() {
    cout.flush();
#ifdef PROCON_LOCAL
    cerr.flush();
    exit(0);
#else
    _Exit(0);
#endif
}

// hash {{{
u64 splitmix64(u64 x) noexcept {
    x += 0x9e3779b97f4a7c15;
    x  = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
    x  = (x ^ (x >> 27)) * 0x94d049bb133111eb;
    return x ^ (x >> 31);
}

u64 RANDOM_SEED() noexcept {
    int dummy;
    static const u64 res =
        splitmix64(chrono::high_resolution_clock::now().time_since_epoch().count()) +
        splitmix64(reinterpret_cast<u64>(&dummy)) +
        splitmix64(reinterpret_cast<u64>(new char));
    return res;
}

template<typename T, typename Enable=void>
struct procon_hash {
    static_assert(!is_pointer<T>::value, "procon_hash<T*> is not supported");
    static_assert(!is_floating_point<T>::value, "procon_hash<Real> is not supported");
    size_t operator()(const T& x) const noexcept {
        return hash<T>{}(x);
    }
};

template<typename T>
size_t procon_hash_value(const T& x) noexcept {
    return procon_hash<T>{}(x);
}

template<typename InputIt>
size_t procon_hash_range(InputIt first, InputIt last) noexcept {
    size_t res = 0;
    for(; first != last; ++first) {
        res *= 2;
        res += procon_hash_value(*first);
    }
    return res;
}

template<typename T>
struct procon_hash<T,enable_if_t<is_integral<T>::value>> {
    size_t operator()(T x) const noexcept {
        return splitmix64(x + RANDOM_SEED());
    }
};

template<>
struct procon_hash<string> {
    static size_t BASE() noexcept {
        return 14695981039346656037ULL + RANDOM_SEED();
    }
    size_t operator()(const string& s) const noexcept {
        static constexpr size_t P = 1099511628211ULL;
        size_t res = BASE();
        for(char c : s) {
            res ^= c;
            res *= P;
        }
        return res;
    }
};

template<typename T1, typename T2>
struct procon_hash<pair<T1,T2>> {
    size_t operator()(const pair<T1,T2>& p) const noexcept {
        size_t h1 = procon_hash_value(FST(p));
        size_t h2 = procon_hash_value(SND(p));
        return 2*h1 + h2;
    }
};

template<typename... TS>
struct procon_hash<tuple<TS...>> {
    size_t operator()(const tuple<TS...>& t) const noexcept {
        size_t res = 0;
        tuple_enumerate(t, [&res](i64, const auto& e) noexcept {
            res *= 2;
            res += procon_hash_value(e);
        });
        return res;
    }
};

template<typename T>
struct procon_hash<vector<T>> {
    size_t operator()(const vector<T>& v) const noexcept {
        return ALL(procon_hash_range, v);
    }
};

template<typename T, typename Hash=procon_hash<T>, typename Eq=equal_to<T>>
using HashSet = unordered_set<T,Hash,Eq>;

template<typename K, typename V, typename Hash=procon_hash<K>, typename Eq=equal_to<K>>
using HashMap = unordered_map<K,V,Hash,Eq>;

template<typename T, typename Hash=procon_hash<T>, typename Eq=equal_to<T>>
using HashMultiset = unordered_multiset<T,Hash,Eq>;

template<typename K, typename V, typename Hash=procon_hash<K>, typename Eq=equal_to<K>>
using HashMultimap = unordered_multimap<K,V,Hash,Eq>;

template<typename T, typename Hash=procon_hash<T>, typename Eq=equal_to<T>>
auto make_hash_set(i64 cap, f32 load_max=0.25) {
    HashSet<T,Hash,Eq> res;
    res.max_load_factor(load_max);
    res.reserve(cap);
    return res;
}

template<typename K, typename V, typename Hash=procon_hash<K>, typename Eq=equal_to<K>>
auto make_hash_map(i64 cap, f32 load_max=0.25) {
    HashMap<K,V,Hash,Eq> res;
    res.max_load_factor(load_max);
    res.reserve(cap);
    return res;
}

template<typename T, typename Hash=procon_hash<T>, typename Eq=equal_to<T>>
auto make_hash_multiset(i64 cap, f32 load_max=0.25) {
    HashMultiset<T,Hash,Eq> res;
    res.max_load_factor(load_max);
    res.reserve(cap);
    return res;
}

template<typename K, typename V, typename Hash=procon_hash<K>, typename Eq=equal_to<K>>
auto make_hash_multimap(i64 cap, f32 load_max=0.25) {
    HashMultimap<K,V,Hash,Eq> res;
    res.max_load_factor(load_max);
    res.reserve(cap);
    return res;
}
// }}}

// stack/queue/priority_queue {{{
template<typename T>
using MaxHeap = priority_queue<T, vector<T>, less<T>>;
template<typename T>
using MinHeap = priority_queue<T, vector<T>, greater<T>>;

template<typename T, typename C>
T POP(stack<T,C>& stk) {
    T x = stk.top(); stk.pop();
    return x;
}

template<typename T, typename C>
T POP(queue<T,C>& que) {
    T x = que.front(); que.pop();
    return x;
}

template<typename T, typename C, typename Comp>
T POP(priority_queue<T,C,Comp>& que) {
    T x = que.top(); que.pop();
    return x;
}
// }}}

// debug {{{
template<typename... TS, SFINAE(sizeof...(TS) == 1)>
void DBG_IMPL(i64 line, const char* expr, const tuple<TS...>& value) {
    cerr << "[L " << line << "]: ";
    cerr << expr << " = ";
    WRITE_REPR(cerr, get<0>(value));
    cerr << "\n";
}

template<typename... TS, SFINAE(sizeof...(TS) >= 2)>
void DBG_IMPL(i64 line, const char* expr, const tuple<TS...>& value) {
    cerr << "[L " << line << "]: ";
    cerr << "(" << expr << ") = ";
    WRITE_REPR(cerr, value);
    cerr << "\n";
}

template<typename T, size_t N>
void DBG_CARRAY_IMPL(i64 line, const char* expr, const T (&ary)[N]) {
    cerr << "[L " << line << "]: ";
    cerr << expr << " = ";
    WRITE_RANGE_REPR(cerr, begin(ary), end(ary));
    cerr << "\n";
}

template<typename InputIt>
void DBG_RANGE_IMPL(i64 line, const char* expr1, const char* expr2, InputIt first, InputIt last) {
    cerr << "[L " << line << "]: ";
    cerr << expr1 << "," << expr2 << " = ";
    WRITE_RANGE_REPR(cerr, first, last);
    cerr << "\n";
}

#ifdef PROCON_LOCAL
    #define DBG(args...) DBG_IMPL(__LINE__, CPP_STR_I(args), std::make_tuple(args))
    #define DBG_CARRAY(expr) DBG_CARRAY_IMPL(__LINE__, CPP_STR(expr), (expr))
    #define DBG_RANGE(first,last) DBG_RANGE_IMPL(__LINE__, CPP_STR(first), CPP_STR(last), (first), (last))
#else
    #define DBG(args...)
    #define DBG_CARRAY(expr)
    #define DBG_RANGE(first,last)
#endif
// }}}

#define PAIR  make_pair
#define TUPLE make_tuple
// }}}

// init {{{
struct ProconInit {
    static constexpr int IOS_PREC = 15;
    static constexpr bool AUTOFLUSH = false;

    ProconInit() {
        cin.tie(nullptr);
        ios::sync_with_stdio(false);
        cin.exceptions(ios::failbit | ios::badbit);
        cout << fixed << setprecision(IOS_PREC);
#ifdef PROCON_LOCAL
        cerr << fixed << setprecision(IOS_PREC);
#endif
        if(AUTOFLUSH)
            cout << unitbuf;
    }
} PROCON_INIT;
// }}}

// num {{{

// 事前条件: a >= 0, b >= 0
i64 extgcd_impl(i64 a, i64 b, i64& x, i64& y) {
    if(b == 0) {
        x = 1; y = 0;
        return a;
    }
    i64 g = extgcd_impl(b, a%b, y, x);
    y -= a/b * x;
    return g;
}

// g=gcd(a,b), および ax+by=g の整数解 (x0,y0) を求める
// (g,x0,y0) を返す
// g!=0 のとき、一般解は (x,y) = (x0+m*b/g, y0-m*a/g) で与えられる(mは整数)
tuple<i64,i64,i64> extgcd(i64 a, i64 b) {
    i64 x, y;
    i64 g = extgcd_impl(ABS(a), ABS(b), x, y);
    x *= sgn(a);
    y *= sgn(b);
    return make_tuple(g, x, y);
}

vector<i64> divisors_proper(i64 n) {
    if(n == 1) return {};
    vector<i64> res(1, 1);

    i64 d = 2;
    for(; d*d < n; ++d) {
        if(n % d == 0) {
            res.emplace_back(d);
            res.emplace_back(n/d);
        }
    }
    if(d*d == n)
        res.emplace_back(d);

    return res;
}

vector<i64> divisors(i64 n) {
    vector<i64> res = divisors_proper(n);
    res.emplace_back(n);
    return res;
}

// 素因数分解
// (素因数,指数) のリストを返す
// n >= 1 でなければならない
// n == 1 の場合、空リストを返す
vector<pair<i64,i64>> factorize(i64 n) {
    ASSERT(n >= 1);

    vector<pair<i64,i64>> res;

    i64 m = n;
    for(i64 i = 2; i*i <= n; ++i) {
        if(m == 1) break;
        i64 e = 0;
        while(m % i == 0) {
            ++e;
            m /= i;
        }
        if(e) res.emplace_back(i, e);
    }
    if(m > 1) res.emplace_back(m, 1);

    return res;
}

bool is_square(i64 x) {
    i64 r = sqrt_floor(x);
    return r*r == x;
}

// Miller-Rabin 法
//
// 参考: http://miller-rabin.appspot.com/
bool is_prime_u32(u32 n) {
    static constexpr u32 AS[] {
         2,
         7,
        61,
    };

    static const auto mulmod32 = [](u32 a, u32 b, u32 m) -> u32 {
        u64 res = a;
        res *= b;
        res %= m;
        return static_cast<u32>(res);
    };

    static const auto powmod32 = [](u32 a, u32 b, u32 m) -> u32 {
        u32 res = 1;
        while(b > 0) {
            if(b & 1)
                res = mulmod32(res, a, m);
            a = mulmod32(a, a, m);
            b >>= 1;
        }
        return res;
    };

    if(n <= 1)     return false;
    if(n == 2)     return true;
    if(n % 2 == 0) return false;

    u32 d = n-1;
    u32 s = __builtin_ctz(d);
    d >>= s;

    for(u32 a : AS) {
        if(a >= n) a %= n;
        if(a == 0) continue;

        u32 x = powmod32(a, d, n);
        if(x == 1 || x == n-1) continue;

        u32 r;
        for(r = 1; r < s; ++r) {
            x = mulmod32(x, x, n);
            if(x == 1)   return false;
            if(x == n-1) break;
        }
        if(r == s) return false;
    }

    return true;
}

bool is_prime_u64(u64 n) {
    static constexpr u64 AS[] {
                 2,
               325,
              9375,
             28178,
            450775,
           9780504,
        1795265022,
    };

    static const auto mulmod64 = [](u64 a, u64 b, u64 m) -> u64 {
        u128 res = a;
        res *= b;
        res %= m;
        return static_cast<u64>(res);
    };

    static const auto powmod64 = [](u64 a, u64 b, u64 m) -> u64 {
        u64 res = 1;
        while(b > 0) {
            if(b & 1)
                res = mulmod64(res, a, m);
            a = mulmod64(a, a, m);
            b >>= 1;
        }
        return res;
    };

    if(n <= numeric_limits<u32>::max()) return is_prime_u32(static_cast<u32>(n));
    if(n % 2 == 0) return false;

    u64 d = n-1;
    u64 s = __builtin_ctzll(d);
    d >>= s;

    for(u64 a : AS) {
        if(a >= n) a %= n;
        if(a == 0) continue;

        u64 x = powmod64(a, d, n);
        if(x == 1 || x == n-1) continue;

        u64 r;
        for(r = 1; r < s; ++r) {
            x = mulmod64(x, x, n);
            if(x == 1)   return false;
            if(x == n-1) break;
        }
        if(r == s) return false;
    }

    return true;
}

bool is_prime(i64 n) {
    ASSERT(n >= 0);
    return is_prime_u64(static_cast<u64>(n));
}

// 二分累乗
template<typename Monoid>
Monoid pow_binary(Monoid x, i64 e) {
    ASSERT(e >= 0);

    Monoid res(1);  // 行列などの場合はここを適当に変える
    Monoid cur = x;
    while(e > 0) {
        if(e & 1)
            res *= cur;
        cur *= cur;
        e >>= 1;
    }
    return res;
}

// mod m での a の逆元
// a ⊥ m でなければならない
i64 inv_mod(i64 a, i64 m) {
    i64 g,x0; tie(g,x0,ignore) = extgcd(a, m);
    ASSERT(g == 1);
    return modulo(x0, m);
}

template<i64 P>
struct ModPT {
    static_assert(P >= 2, "P must be a prime");
    i64 v_;  // [0,P)

    ModPT() : v_(0) {}
    ModPT(i64 v) : v_(modulo(v,P)) {}

    ModPT operator-() const {
        return ModPT(-v_);
    }
    ModPT& operator+=(ModPT rhs) {
        v_ += rhs.v_;
        v_ %= P;
        return *this;
    }
    ModPT& operator-=(ModPT rhs) {
        v_ += P;
        v_ -= rhs.v_;
        v_ %= P;
        return *this;
    }
    ModPT& operator*=(ModPT rhs) {
        v_ *= rhs.v_;
        v_ %= P;
        return *this;
    }

    ModPT& operator++() {
        return *this += 1;
    }
    ModPT& operator--() {
        return *this -= 1;
    }
    ModPT operator++(int) {
        return exchange(*this, *this+1);
    }
    ModPT operator--(int) {
        return exchange(*this, *this-1);
    }

    explicit operator i64() const { return v_; }

    ModPT inv() const {
        return ModPT(inv_mod(v_,P));
    }
};

template<i64 P>
ModPT<P> operator+(ModPT<P> lhs, ModPT<P> rhs) { return ModPT<P>(lhs) += rhs; }
template<i64 P>
ModPT<P> operator+(ModPT<P> lhs, i64 rhs) { return ModPT<P>(lhs) += rhs; }
template<i64 P>
ModPT<P> operator+(i64 lhs, ModPT<P> rhs) { return ModPT<P>(rhs) += lhs; }
template<i64 P>
ModPT<P> operator-(ModPT<P> lhs, ModPT<P> rhs) { return ModPT<P>(lhs) -= rhs; }
template<i64 P>
ModPT<P> operator-(ModPT<P> lhs, i64 rhs) { return ModPT<P>(lhs) -= rhs; }
template<i64 P>
ModPT<P> operator-(i64 lhs, ModPT<P> rhs) { return ModPT<P>(rhs) -= lhs; }
template<i64 P>
ModPT<P> operator*(ModPT<P> lhs, ModPT<P> rhs) { return ModPT<P>(lhs) *= rhs; }
template<i64 P>
ModPT<P> operator*(ModPT<P> lhs, i64 rhs) { return ModPT<P>(lhs) *= rhs; }
template<i64 P>
ModPT<P> operator*(i64 lhs, ModPT<P> rhs) { return ModPT<P>(rhs) *= lhs; }

template<i64 P>
bool operator==(ModPT<P> lhs, ModPT<P> rhs) { return lhs.v_ == rhs.v_; }
template<i64 P>
bool operator==(ModPT<P> lhs, i64 rhs) { return lhs == ModPT<P>(rhs); }
template<i64 P>
bool operator==(i64 lhs, ModPT<P> rhs) { return ModPT<P>(lhs) == rhs; }
template<i64 P>
bool operator!=(ModPT<P> lhs, ModPT<P> rhs) { return !(lhs == rhs); }
template<i64 P>
bool operator!=(ModPT<P> lhs, i64 rhs) { return !(lhs == rhs); }
template<i64 P>
bool operator!=(i64 lhs, ModPT<P> rhs) { return !(lhs == rhs); }

template<i64 P>
struct Scanner<ModPT<P>> {
    static ModPT<P> read(istream& in) {
        return READ<i64>(in);
    }
};

template<i64 P>
struct Formatter<ModPT<P>> {
    static ostream& write_str(ostream& out, ModPT<P> x) {
        return WRITE_STR(out, x.v_);
    }
    static ostream& write_repr(ostream& out, ModPT<P> x) {
        return WRITE_REPR(out, x.v_);
    }
};

using ModP = ModPT<MOD>;

// エラトステネスのふるい
template<i64 N>
bool (&is_prime_table())[N] {
    static_assert(N >= 3, "");
    static bool prime[N] {};

    if(!prime[2]) {
        fill(begin(prime)+2, end(prime), true);
        for(i64 i = 2; i*i <= N-1; ++i) {
            if(!prime[i]) continue;
            for(i64 j = i+i; j < N; j += i)
                prime[j] = false;
        }
    }
    return prime;
}

// F(0) = 0
// F(1) = 1
// F(n) = F(n-1) + F(n-2)
//
// // decltype(auto) で受けると SIZE() が使える (auto だとポインタになってしまう)
// decltype(auto) fib = fibonacci_table<1000>();
template<i64 N>
ModP (&fibonacci_table())[N] {
    static_assert(N >= 2, "");
    static ModP fib[N] {};

    if(fib[1] != 1) {
        fib[0] = 0;
        fib[1] = 1;
        FOR(i, 2, N) {
            fib[i] = fib[i-1] + fib[i-2];
        }
    }
    return fib;
}

template<i64 N>
ModP (&factorial_table())[N] {
    static_assert(N >= 1, "");
    static ModP fac[N] {};

    if(fac[0] != 1) {
        fac[0] = 1;
        FOR(i, 1, N) {
            fac[i] = i * fac[i-1];
        }
    }
    return fac;
}

template<i64 N>
ModP (&ifactorial_table())[N] {
    static_assert(N >= 1, "");
    static ModP ifac[N] {};

    if(ifac[0] != 1) {
        decltype(auto) fac = factorial_table<N>();
        ifac[N-1] = fac[N-1].inv();
        for(i64 i = N-2; i >= 0; --i) {
            ifac[i] = (i+1) * ifac[i+1];
        }
    }
    return ifac;
}

ModP permutation_count_fac(i64 n, i64 r, const ModP* fac, const ModP* ifac) {
    if(n < r) return 0;
    return fac[n] * ifac[n-r];
}

template<i64 H, i64 W>
ModP (&combination_count_table())[H][W] {
    static_assert(W >= 1 && H >= W, "");
    static ModP dp[H][W] {};

    if(dp[0][0] != 1) {
        REP(i, H) {
            dp[i][0] = 1;
            dp[i][i] = 1;
        }
        FOR(i, 1, H) FOR(j, 1, i) {
            dp[i][j] = dp[i-1][j-1] + dp[i-1][j];
        }
    }
    return dp;
}

template<i64 H, i64 W>
auto combination_count_func() {
    static_assert(W >= 1 && H >= W, "");
    return MEMOIZE<H,W>([](auto&& self, i64 n, i64 r) -> ModP {
        if(n <  r) return 0;
        if(r == 0) return 1;
        if(n == r) return 1;
        return self(n-1,r-1) + self(n-1,r);
    });
}

ModP combination_count_fac(i64 n, i64 r, const ModP* fac, const ModP* ifac) {
    if(n < r) return 0;
    return fac[n] * ifac[r] * ifac[n-r];
}

// 分割数 P(n,k) (n を k 個の正整数の和で表す場合の数)
//
// 「n を 最大値 k の正整数の和で表す場合の数」でもある。
// 「n を k 個『以下』の正整数の和で表す場合の数」は sum(P(n,i)) (1<=i<=k)
// 「n を k 個の『非負整数』の和で表す場合の数」は P(n+k,k)
//
// P(0,0) = 1
// P(n,0) = 0
// P(0,k) = 0
// n < k のとき P(n,k) = 0
// P(n,1) = 1
// P(n,n) = 1
//
// 順序を区別するものは合成(composition)とよばれる。
// 「n を k 個の順序つき正整数列の和で表す場合の数」は Q(n,k) = comb(n-1,k-1)
// 「n を k 個の順序つき『非負整数』列の和で表す場合の数」は Q(n+k,k) = comb(n+k-1,k-1) = comb(n+k-1,n)
// 「n を順序つき正整数列の和で表す場合の数」は 2^(n-1)
template<i64 H, i64 W>
ModP (&partition_count_table())[H][W] {
    static_assert(W >= 1 && H >= W, "");
    static ModP dp[H][W] {};

    if(dp[0][0] != 1) {
        REP(j, W) {
            dp[j][j] = 1;
        }
        FOR(i, 2, H) {
            dp[i][1] = 1;
        }
        FOR(i, 3, H) {
            FOR(j, 2, MIN(i,W)) {
                dp[i][j] = dp[i-1][j-1] + dp[i-j][j];
            }
        }
    }
    return dp;
}

// 分割数 メモ化再帰版
template<i64 H, i64 W>
auto partition_count_func() {
    static_assert(W >= 1 && H >= W, "");
    return MEMOIZE<H,W>([](auto&& self, i64 n, i64 k) -> ModP {
        if(n <  k) return 0;
        if(n == k) return 1;
        if(k == 1) return 1;
        return self(n-1,k-1) + self(n-k,k);
    });
}

// }}}

// container {{{

template<typename T, typename Comp>
auto make_priority_queue(const Comp& comp) {
    return priority_queue<T,vector<T>,Comp>(comp);
}

template<typename T, typename InputIt, typename Comp>
auto make_priority_queue(InputIt first, InputIt last, const Comp& comp) {
    return priority_queue<T,vector<T>,Comp>(first, last, comp);
}

// set/map/multiset/multimap search {{{
// set {{{
template<typename T, typename Comp>
auto set_search_lt(set<T,Comp>& s, const typename set<T,Comp>::key_type& x) {
    auto it = s.lower_bound(x);
    if(it == begin(s)) return end(s);
    return prev(it);
}

template<typename T, typename Comp>
auto set_search_lt(const set<T,Comp>& s, const typename set<T,Comp>::key_type& x) {
    auto it = s.lower_bound(x);
    if(it == begin(s)) return end(s);
    return prev(it);
}

template<typename T, typename Comp>
auto set_search_le(set<T,Comp>& s, const typename set<T,Comp>::key_type& x) {
    auto it = s.upper_bound(x);
    if(it == begin(s)) return end(s);
    return prev(it);
}

template<typename T, typename Comp>
auto set_search_le(const set<T,Comp>& s, const typename set<T,Comp>::key_type& x) {
    auto it = s.upper_bound(x);
    if(it == begin(s)) return end(s);
    return prev(it);
}

template<typename T, typename Comp>
auto set_search_gt(set<T,Comp>& s, const typename set<T,Comp>::key_type& x) {
    return s.upper_bound(x);
}

template<typename T, typename Comp>
auto set_search_gt(const set<T,Comp>& s, const typename set<T,Comp>::key_type& x) {
    return s.upper_bound(x);
}

template<typename T, typename Comp>
auto set_search_ge(set<T,Comp>& s, const typename set<T,Comp>::key_type& x) {
    return s.lower_bound(x);
}

template<typename T, typename Comp>
auto set_search_ge(const set<T,Comp>& s, const typename set<T,Comp>::key_type& x) {
    return s.lower_bound(x);
}
// }}}
// map {{{
template<typename K, typename V, typename Comp>
auto map_search_lt(map<K,V,Comp>& m, const typename map<K,V,Comp>::key_type& x) {
    auto it = m.lower_bound(x);
    if(it == begin(m)) return end(m);
    return prev(it);
}

template<typename K, typename V, typename Comp>
auto map_search_lt(const map<K,V,Comp>& m, const typename map<K,V,Comp>::key_type& x) {
    auto it = m.lower_bound(x);
    if(it == begin(m)) return end(m);
    return prev(it);
}

template<typename K, typename V, typename Comp>
auto map_search_le(map<K,V,Comp>& m, const typename map<K,V,Comp>::key_type& x) {
    auto it = m.upper_bound(x);
    if(it == begin(m)) return end(m);
    return prev(it);
}

template<typename K, typename V, typename Comp>
auto map_search_le(const map<K,V,Comp>& m, const typename map<K,V,Comp>::key_type& x) {
    auto it = m.upper_bound(x);
    if(it == begin(m)) return end(m);
    return prev(it);
}

template<typename K, typename V, typename Comp>
auto map_search_gt(map<K,V,Comp>& m, const typename map<K,V,Comp>::key_type& x) {
    return m.upper_bound(x);
}

template<typename K, typename V, typename Comp>
auto map_search_gt(const map<K,V,Comp>& m, const typename map<K,V,Comp>::key_type& x) {
    return m.upper_bound(x);
}

template<typename K, typename V, typename Comp>
auto map_search_ge(map<K,V,Comp>& m, const typename map<K,V,Comp>::key_type& x) {
    return m.lower_bound(x);
}

template<typename K, typename V, typename Comp>
auto map_search_ge(const map<K,V,Comp>& m, const typename map<K,V,Comp>::key_type& x) {
    return m.lower_bound(x);
}
// }}}
// multiset {{{
// 等価な値が複数ある場合、lt/le は先頭を、gt/ge は末尾を指す
template<typename T, typename Comp>
auto set_search_lt(multiset<T,Comp>& s, const typename multiset<T,Comp>::key_type& x) {
    auto it = s.lower_bound(x);
    if(it == begin(s)) return end(s);
    return prev(it);
}

template<typename T, typename Comp>
auto set_search_lt(const multiset<T,Comp>& s, const typename multiset<T,Comp>::key_type& x) {
    auto it = s.lower_bound(x);
    if(it == begin(s)) return end(s);
    return prev(it);
}

template<typename T, typename Comp>
auto set_search_le(multiset<T,Comp>& s, const typename multiset<T,Comp>::key_type& x) {
    auto it = s.upper_bound(x);
    if(it == begin(s)) return end(s);
    return prev(it);
}

template<typename T, typename Comp>
auto set_search_le(const multiset<T,Comp>& s, const typename multiset<T,Comp>::key_type& x) {
    auto it = s.upper_bound(x);
    if(it == begin(s)) return end(s);
    return prev(it);
}

template<typename T, typename Comp>
auto set_search_gt(multiset<T,Comp>& s, const typename multiset<T,Comp>::key_type& x) {
    return s.upper_bound(x);
}

template<typename T, typename Comp>
auto set_search_gt(const multiset<T,Comp>& s, const typename multiset<T,Comp>::key_type& x) {
    return s.upper_bound(x);
}

template<typename T, typename Comp>
auto set_search_ge(multiset<T,Comp>& s, const typename multiset<T,Comp>::key_type& x) {
    return s.lower_bound(x);
}

template<typename T, typename Comp>
auto set_search_ge(const multiset<T,Comp>& s, const typename multiset<T,Comp>::key_type& x) {
    return s.lower_bound(x);
}
// }}}
// multimap {{{
// 等価な値が複数ある場合、lt/le は先頭を、gt/ge は末尾を指す
template<typename K, typename V, typename Comp>
auto map_search_lt(multimap<K,V,Comp>& m, const typename multimap<K,V,Comp>::key_type& x) {
    auto it = m.lower_bound(x);
    if(it == begin(m)) return end(m);
    return prev(it);
}

template<typename K, typename V, typename Comp>
auto map_search_lt(const multimap<K,V,Comp>& m, const typename multimap<K,V,Comp>::key_type& x) {
    auto it = m.lower_bound(x);
    if(it == begin(m)) return end(m);
    return prev(it);
}

template<typename K, typename V, typename Comp>
auto map_search_le(multimap<K,V,Comp>& m, const typename multimap<K,V,Comp>::key_type& x) {
    auto it = m.upper_bound(x);
    if(it == begin(m)) return end(m);
    return prev(it);
}

template<typename K, typename V, typename Comp>
auto map_search_le(const multimap<K,V,Comp>& m, const typename multimap<K,V,Comp>::key_type& x) {
    auto it = m.upper_bound(x);
    if(it == begin(m)) return end(m);
    return prev(it);
}

template<typename K, typename V, typename Comp>
auto map_search_gt(multimap<K,V,Comp>& m, const typename multimap<K,V,Comp>::key_type& x) {
    return m.upper_bound(x);
}

template<typename K, typename V, typename Comp>
auto map_search_gt(const multimap<K,V,Comp>& m, const typename multimap<K,V,Comp>::key_type& x) {
    return m.upper_bound(x);
}

template<typename K, typename V, typename Comp>
auto map_search_ge(multimap<K,V,Comp>& m, const typename multimap<K,V,Comp>::key_type& x) {
    return m.lower_bound(x);
}

template<typename K, typename V, typename Comp>
auto map_search_ge(const multimap<K,V,Comp>& m, const typename multimap<K,V,Comp>::key_type& x) {
    return m.lower_bound(x);
}
// }}}
// }}}

template<typename T, typename Comp>
bool set_contains(const set<T,Comp>& s, const typename set<T,Comp>::key_type& x) {
    return s.find(x) != end(s);
}

template<typename T, typename Hash, typename Eq>
bool set_contains(const unordered_set<T,Hash,Eq>& s, const typename unordered_set<T,Hash,Eq>::key_type& x) {
    return s.find(x) != end(s);
}

template<typename T, typename Comp>
bool set_contains(const multiset<T,Comp>& s, const typename multiset<T,Comp>::key_type& x) {
    return s.find(x) != end(s);
}

template<typename T, typename Hash, typename Eq>
bool set_contains(const unordered_multiset<T,Hash,Eq>& s, const typename unordered_multiset<T,Hash,Eq>::key_type& x) {
    return s.find(x) != end(s);
}

template<typename K, typename V, typename Comp>
bool map_contains(const map<K,V,Comp>& m, const typename map<K,V,Comp>::key_type& k) {
    return m.find(k) != end(m);
}

template<typename K, typename V, typename Hash, typename Eq>
bool map_contains(const unordered_map<K,V,Hash,Eq>& m, const typename unordered_map<K,V,Hash,Eq>::key_type& k) {
    return m.find(k) != end(m);
}

template<typename K, typename V, typename Comp>
bool map_contains(const multimap<K,V,Comp>& m, const typename map<K,V,Comp>::key_type& k) {
    return m.find(k) != end(m);
}

template<typename K, typename V, typename Hash, typename Eq>
bool map_contains(const unordered_multimap<K,V,Hash,Eq>& m, const typename unordered_map<K,V,Hash,Eq>::key_type& k) {
    return m.find(k) != end(m);
}

template<typename K, typename V, typename Comp>
V map_get(const map<K,V,Comp>& m,
          const typename map<K,V,Comp>::key_type& k,
          const typename map<K,V,Comp>::mapped_type& def)
{
    auto it = m.find(k);
    return it == end(m) ? def : it->second;
}

template<typename K, typename V, typename Hash, typename Eq>
V map_get(const unordered_map<K,V,Hash,Eq>& m,
          const typename unordered_map<K,V,Hash,Eq>::key_type& k,
          const typename unordered_map<K,V,Hash,Eq>::mapped_type& def)
{
    auto it = m.find(k);
    return it == end(m) ? def : it->second;
}

template<typename K, typename V, typename Comp>
V& map_setdefault(map<K,V,Comp>& m,
                  const typename map<K,V,Comp>::key_type& k,
                  const typename map<K,V,Comp>::mapped_type& def)
{
    auto it = m.find(k);
    if(it == end(m))
        it = m.emplace_hint(it, k, def);
    return it->second;
}

template<typename K, typename V, typename Hash, typename Eq>
V& map_setdefault(unordered_map<K,V,Hash,Eq>& m,
                  const typename unordered_map<K,V,Hash,Eq>::key_type& k,
                  const typename unordered_map<K,V,Hash,Eq>::mapped_type& def)
{
    auto it = m.find(k);
    if(it == end(m))
        it = m.emplace_hint(it, k, def);
    return it->second;
}

template<typename K, typename V, typename Comp, typename F>
V& map_setdefault_with(map<K,V,Comp>& m,
                       const typename map<K,V,Comp>::key_type& k,
                       F&& f)
{
    auto it = m.find(k);
    if(it == end(m))
        it = m.emplace_hint(it, k, f());
    return it->second;
}

template<typename K, typename V, typename Hash, typename Eq, typename F>
V& map_setdefault_with(unordered_map<K,V,Hash,Eq>& m,
                       const typename unordered_map<K,V,Hash,Eq>::key_type& k,
                       F&& f)
{
    auto it = m.find(k);
    if(it == end(m))
        it = m.emplace_hint(it, k, f());
    return it->second;
}

template<typename K, typename Comp>
bool multiset_erase_one(multiset<K,Comp>& m, const typename multiset<K,Comp>::key_type& k) {
    auto it = m.find(k);
    if(it == end(m)) return false;
    m.erase(it);
    return true;
}

template<typename K, typename Hash, typename Eq>
bool multiset_erase_one(unordered_multiset<K,Hash,Eq>& m, const typename unordered_multiset<K,Hash,Eq>::key_type& k) {
    auto it = m.find(k);
    if(it == end(m)) return false;
    m.erase(it);
    return true;
}

// POP() 系 {{{
// 効率は悪い
template<typename T>
T POP_FRONT(vector<T>& v) {
    T x = v.front(); v.erase(begin(v));
    return x;
}

template<typename T>
T POP_BACK(vector<T>& v) {
    T x = v.back(); v.pop_back();
    return x;
}

template<typename T>
T POP_FRONT(deque<T>& v) {
    T x = v.front(); v.pop_front();
    return x;
}

template<typename T>
T POP_BACK(deque<T>& v) {
    T x = v.back(); v.pop_back();
    return x;
}

template<typename T>
T POP_FRONT(forward_list<T>& ls) {
    T x = ls.front(); ls.pop_front();
    return x;
}

template<typename T>
T POP_FRONT(list<T>& ls) {
    T x = ls.front(); ls.pop_front();
    return x;
}

template<typename T>
T POP_BACK(list<T>& ls) {
    T x = ls.back(); ls.pop_back();
    return x;
}
// }}}

// bimap {{{
template<typename T1, typename T2>
struct BiHashMap {
    HashMap<T1,T2> fwd_;
    HashMap<T2,T1> rev_;

    void insert(const T1& x, const T2& y) {
        auto it_fwd = fwd_.find(x);
        if(it_fwd == end(fwd_)) {
            fwd_.insert(it_fwd, make_pair(x,y));
            rev_.insert(end(rev_), make_pair(y,x));
        }
        else {
            ASSERT(y == it_fwd->second);
        }
    }

    bool contains_fwd(const T1& x) const {
        return map_contains(fwd_, x);
    }

    bool contains_rev(const T2& y) const {
        return map_contains(rev_, y);
    }

    const T2& at_fwd(const T1& x) const {
        auto it = fwd_.find(x);
        ASSERT(it != end(fwd_));
        return it->second;
    }

    const T1& at_rev(const T2& y) const {
        auto it = rev_.find(y);
        ASSERT(it != end(rev_));
        return it->second;
    }

    size_t size() const { return fwd_.size(); }
};
// }}}

// Formatter {{{
template<typename T, size_t N>
struct Formatter<array<T,N>> {
    static ostream& write_str(ostream& out, const array<T,N>& a) {
        return WRITE_RANGE_STR(out, begin(a), end(a));
    }
    static ostream& write_repr(ostream& out, const array<T,N>& a) {
        out << "array";
        return WRITE_RANGE_REPR(out, begin(a), end(a));
    }
};

template<typename T>
struct Formatter<deque<T>> {
    static ostream& write_str(ostream& out, const deque<T>& deq) {
        return WRITE_RANGE_STR(out, begin(deq), end(deq));
    }
    static ostream& write_repr(ostream& out, const deque<T>& deq) {
        out << "deque";
        return WRITE_RANGE_REPR(out, begin(deq), end(deq));
    }
};

template<typename T>
struct Formatter<forward_list<T>> {
    static ostream& write_str(ostream& out, const forward_list<T>& ls) {
        return WRITE_RANGE_STR(out, begin(ls), end(ls));
    }
    static ostream& write_repr(ostream& out, const forward_list<T>& ls) {
        out << "forward_list";
        return WRITE_RANGE_REPR(out, begin(ls), end(ls));
    }
};

template<typename T>
struct Formatter<list<T>> {
    static ostream& write_str(ostream& out, const list<T>& ls) {
        return WRITE_RANGE_STR(out, begin(ls), end(ls));
    }
    static ostream& write_repr(ostream& out, const list<T>& ls) {
        out << "list";
        return WRITE_RANGE_REPR(out, begin(ls), end(ls));
    }
};

template<typename T, typename Comp>
struct Formatter<set<T,Comp>> {
    static ostream& write_str(ostream& out, const set<T,Comp>& s) {
        return WRITE_RANGE_STR(out, begin(s), end(s));
    }
    static ostream& write_repr(ostream& out, const set<T,Comp>& s) {
        out << "set";
        return WRITE_RANGE_REPR(out, begin(s), end(s));
    }
};

template<typename T, typename Comp>
struct Formatter<multiset<T,Comp>> {
    static ostream& write_str(ostream& out, const multiset<T,Comp>& s) {
        return WRITE_RANGE_STR(out, begin(s), end(s));
    }
    static ostream& write_repr(ostream& out, const multiset<T,Comp>& s) {
        out << "multiset";
        return WRITE_RANGE_REPR(out, begin(s), end(s));
    }
};

template<typename T, typename Hash, typename Eq>
struct Formatter<unordered_set<T,Hash,Eq>> {
    static ostream& write_str(ostream& out, const unordered_set<T,Hash,Eq>& s) {
        return WRITE_RANGE_STR(out, begin(s), end(s));
    }
    static ostream& write_repr(ostream& out, const unordered_set<T,Hash,Eq>& s) {
        out << "unordered_set";
        return WRITE_RANGE_REPR(out, begin(s), end(s));
    }
};

template<typename T, typename Hash, typename Eq>
struct Formatter<unordered_multiset<T,Hash,Eq>> {
    static ostream& write_str(ostream& out, const unordered_multiset<T,Hash,Eq>& s) {
        return WRITE_RANGE_STR(out, begin(s), end(s));
    }
    static ostream& write_repr(ostream& out, const unordered_multiset<T,Hash,Eq>& s) {
        out << "unordered_multiset";
        return WRITE_RANGE_REPR(out, begin(s), end(s));
    }
};

template<typename K, typename V, typename Comp>
struct Formatter<map<K,V,Comp>> {
    static ostream& write_str(ostream& out, const map<K,V,Comp>& m) {
        return WRITE_RANGE_STR(out, begin(m), end(m));
    }
    static ostream& write_repr(ostream& out, const map<K,V,Comp>& m) {
        out << "map";
        return WRITE_RANGE_REPR(out, begin(m), end(m));
    }
};

template<typename K, typename V, typename Comp>
struct Formatter<multimap<K,V,Comp>> {
    static ostream& write_str(ostream& out, const multimap<K,V,Comp>& m) {
        return WRITE_RANGE_STR(out, begin(m), end(m));
    }
    static ostream& write_repr(ostream& out, const multimap<K,V,Comp>& m) {
        out << "multimap";
        return WRITE_RANGE_REPR(out, begin(m), end(m));
    }
};

template<typename K, typename V, typename Hash, typename Eq>
struct Formatter<unordered_map<K,V,Hash,Eq>> {
    static ostream& write_str(ostream& out, const unordered_map<K,V,Hash,Eq>& m) {
        return WRITE_RANGE_STR(out, begin(m), end(m));
    }
    static ostream& write_repr(ostream& out, const unordered_map<K,V,Hash,Eq>& m) {
        out << "unordered_map";
        return WRITE_RANGE_REPR(out, begin(m), end(m));
    }
};

template<typename K, typename V, typename Hash, typename Eq>
struct Formatter<unordered_multimap<K,V,Hash,Eq>> {
    static ostream& write_str(ostream& out, const unordered_multimap<K,V,Hash,Eq>& m) {
        return WRITE_RANGE_STR(out, begin(m), end(m));
    }
    static ostream& write_repr(ostream& out, const unordered_multimap<K,V,Hash,Eq>& m) {
        out << "unordered_multimap";
        return WRITE_RANGE_REPR(out, begin(m), end(m));
    }
};

template<typename T, typename C>
struct Formatter<stack<T,C>> {
    static ostream& write_str(ostream& out, const stack<T,C>& orig) {
        stack<T,C> stk(orig);
        while(!stk.empty()) {
            WRITE_STR(out, stk.top()); stk.pop();
            if(!stk.empty()) out << ' ';
        }
        return out;
    }
    static ostream& write_repr(ostream& out, const stack<T,C>& orig) {
        stack<T,C> stk(orig);
        out << "stack[";
        while(!stk.empty()) {
            WRITE_REPR(out, stk.top()); stk.pop();
            if(!stk.empty()) out << ", ";
        }
        out << "]";
        return out;
    }
};

template<typename T, typename C>
struct Formatter<queue<T,C>> {
    static ostream& write_str(ostream& out, const queue<T,C>& orig) {
        queue<T,C> que(orig);
        while(!que.empty()) {
            WRITE_STR(out, que.front()); que.pop();
            if(!que.empty()) out << ' ';
        }
        return out;
    }
    static ostream& write_repr(ostream& out, const queue<T,C>& orig) {
        queue<T,C> que(orig);
        out << "queue[";
        while(!que.empty()) {
            WRITE_REPR(out, que.front()); que.pop();
            if(!que.empty()) out << ", ";
        }
        out << "]";
        return out;
    }
};

template<typename T, typename C, typename Comp>
struct Formatter<priority_queue<T,C,Comp>> {
    static ostream& write_str(ostream& out, const priority_queue<T,C,Comp>& orig) {
        priority_queue<T,C,Comp> que(orig);
        while(!que.empty()) {
            WRITE_STR(out, que.top()); que.pop();
            if(!que.empty()) out << ' ';
        }
        return out;
    }
    static ostream& write_repr(ostream& out, const priority_queue<T,C,Comp>& orig) {
        priority_queue<T,C,Comp> que(orig);
        out << "priority_queue[";
        while(!que.empty()) {
            WRITE_REPR(out, que.top()); que.pop();
            if(!que.empty()) out << ", ";
        }
        out << "]";
        return out;
    }
};
// }}}

// }}}

//--------------------------------------------------------------------



void solve() {
    i64 N = RD();
    i64 K = RD();
    auto A = RD_ARRAY(N);

    i64 sum  = ALL(SUM, A);
#if 0
    i64 amax = *ALL(max_element, A);
    DBG(sum,amax);
    if(K >= sum-amax) {
        PRINTLN(sum);
        return;
    }
#endif

    auto ds = divisors(sum);

    auto is_ok = [K,&A](i64 d) -> bool {
        deque<i64> deq;
        ALL(transform, A, back_inserter(deq), [d](i64 a) { return a%d; });
        ALL(sort, deq);
        DBG(deq);

        i64 remain = K;
        DBG(remain);
        while(SIZE(deq) >= 2) {
            i64 cost1 = deq.front();
            i64 cost2 = d-deq.back();
            DBG(cost1,cost2);
            if(cost1 <= cost2) {
                remain -= cost1;
                deq.back() += cost1;
                deq.pop_front();
            }
            else {
                remain -= cost2;
                deq.front() -= cost2;
                deq.pop_back();
            }
            DBG(d,deq,remain);
            if(remain < 0) return false;
        }
        DBG(d,deq);
        return deq.front()%d == 0;
    };

    i64 ans = 1;
    for(i64 d : ds) {
        if(is_ok(d))
            chmax(ans, d);
    }

    PRINTLN(ans);

    // * 小さいケースで試した?
    // * 不可能なケースはチェックした?
    // * MOD はとった?
    // * メモ化忘れてない?
    // * 入出力の 0-based/1-based 確認した?
    // * 時間/メモリ制限は確認した?
    // * 違うやつ提出してない?
    // * 違うやつテストしてない?
}

signed main() {
    

    solve();

    EXIT();
}


Measured memory_bytes: 256