/**
 * 
 */

#define ASSERT_LV 1
// header {{{
#ifndef ASSERT_LV
# define ASSERT_LV 1
#endif
#if ASSERT_LV == 0
# define NDEBUG
#endif

#if defined(__GNUC__) && !defined(__clang__)
#include <bits/stdc++.h>
#else
#include <cassert>
#include <cctype>
#include <cerrno>
#include <cfloat>
#include <ciso646>
#include <climits>
//#include <clocale>
#include <cmath>
//#include <csetjmp>
//#include <csignal>
#include <cstdarg>
#include <cstddef>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <ctime>
//#include <cwchar>
//#include <cwctype>
#if __cplusplus >= 201103L
//#include <ccomplex>
#include <cfenv>
#include <cinttypes>
//#include <cstdalign>
//#include <cstdbool>
#include <cstdint>
//#include <ctgmath>
//#include <cuchar>
#endif
#include <algorithm>
#include <bitset>
#include <complex>
#include <deque>
#include <exception>
#include <fstream>
#include <functional>
#include <iomanip>
#include <ios>
#include <iosfwd>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
//#include <locale>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <set>
#include <sstream>
#include <stack>
#include <stdexcept>
#include <streambuf>
#include <string>
#include <typeinfo>
#include <utility>
#include <valarray>
#include <vector>
#if __cplusplus >= 201103L
#include <array>
//#include <atomic>
#include <chrono>
//#include <codecvt>
//#include <condition_variable>
#include <forward_list>
//#include <future>
#include <initializer_list>
//#include <mutex>
#include <random>
#include <ratio>
#include <regex>
#include <scoped_allocator>
//#include <system_error>
#include <thread>
#include <tuple>
#include <typeindex>
#include <type_traits>
#include <unordered_map>
#include <unordered_set>
#endif
#if __cplusplus >= 201402L
//#include <shared_mutex>
#endif
#if __cplusplus >= 201703L
#include <any>
//#include <charconv>
//#include <execution>
//#include <filesystem>
#include <optional>
//#include <memory_resource>
#include <string_view>
#include <variant>
#endif
#endif
using namespace std;

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 = long double;

template<class T> constexpr T PROCON_INF();
template<> constexpr i32 PROCON_INF<i32>() { return 1'010'000'011; }
template<> constexpr i64 PROCON_INF<i64>() { return INT64_C(1'010'000'000'000'000'017); }
template<> constexpr f32 PROCON_INF<f32>() { return 1e19F; }
template<> constexpr f64 PROCON_INF<f64>() { return 1e100; }
template<> constexpr f80 PROCON_INF<f80>() { return 1e100L; }
// }}}

using Int  = i64;
using Real = f80;

constexpr Int MOD = 1'000'000'007;
//constexpr Int MOD = 998'244'353;

constexpr Real EPS = Real(1e-10L);

constexpr int  COUT_PREC      = 15;
constexpr bool COUT_AUTOFLUSH = false;

// procon {{{

static_assert(is_same<Int,i64>::value || is_same<Int,i32>::value, "");
static_assert(is_same<Real,f80>::value || is_same<Real,f64>::value || is_same<Real,f32>::value, "");

#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))
#if defined(PROCON_LOCAL) || ASSERT_LV >= 2
# define ASSERT_LOCAL(expr...) assert((expr))
#else
# define ASSERT_LOCAL(expr...)
#endif

constexpr Int  INF  = PROCON_INF<Int>();
constexpr Real FINF = PROCON_INF<Real>();

constexpr Real PI = Real(3.141592653589793238462643383279502884197L);

template<class T> constexpr T SQRT_MAX();
template<> constexpr i32 SQRT_MAX<i32>() { return 46340; }
template<> constexpr i64 SQRT_MAX<i64>() { return INT64_C(3037000499); }

template<class T, SFINAE(is_signed<T>::value)>
constexpr T ABS(T x) noexcept {
    return x < 0 ? -x : x;
}

constexpr bool LT_EPS(Real lhs, Real rhs, Real eps=EPS) { return lhs < rhs-eps; }
constexpr bool GT_EPS(Real lhs, Real rhs, Real eps=EPS) { return lhs > rhs+eps; }
constexpr bool EQ_EPS(Real lhs, Real rhs, Real eps=EPS) { return ABS(lhs-rhs) <= eps; }

constexpr bool EQ_EXACT(Real lhs, Real rhs) {
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
    return lhs == rhs;
#pragma GCC diagnostic pop
}

#define FOR(i, start, end) for(Int i = (start), CPP_CAT(i,xxxx_end)=(end); i < CPP_CAT(i,xxxx_end); ++i)
#define REP(i, n) FOR(i, 0, n)
#define LOOP(n) REP(CPP_CAT(macro_loop_counter,__COUNTER__), 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 LIFT(f) ([](auto&&... args) -> decltype(auto) { return (f)(std::forward<decltype(args)>(args)...); })

template<class C>
constexpr Int SIZE(const C& c) noexcept { return Int(c.size()); }

template<class T, size_t N>
constexpr Int SIZE(const T (&)[N]) noexcept { return Int(N); }

constexpr bool is_odd (Int x) { return x%2 != 0; }
constexpr bool is_even(Int x) { return x%2 == 0; }
constexpr Int PARITY(Int x) { return x%2==0 ? 0 : 1; }

template<class T>
constexpr Int CMP(T x, T y) noexcept { return (y<x) - (x<y); }

template<class T>
constexpr Int SGN(T x) noexcept { return CMP(x,T(0)); }

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

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

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

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

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

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

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

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

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

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

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

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

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

template<class F>
auto NOT_FN(F&& f) {
    return [f=forward<F>(f)](auto&&... args) -> bool { return !f(forward<decltype(args)>(args)...); };
}

struct IDENTITY {
    using is_transparent = void;

    template<class T>
    constexpr decltype(auto) operator()(T&& x) const { return forward<T>(x); }
};

// ビット演算 {{{
// 引数は [-INF,INF] のみ想定
constexpr Int BIT_I(Int i) {
    return Int(1) << i;
}

constexpr Int BIT_I_1(Int i) {
    return BIT_I(i) - 1;
}

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

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

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

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

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

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

/*constexpr*/ Int BIT_COUNT_LEADING_ZEROS(Int x) {
    if(is_same<Int,i64>::value)
        return x==0 ? 64 : __builtin_clzll(u64(x));
    else if(is_same<Int,i32>::value)
        return x==0 ? 32 : __builtin_clz(u32(x));
    ASSERT(false);
}

/*constexpr*/ Int BIT_COUNT_TRAILING_ZEROS(Int x) {
    if(is_same<Int,i64>::value)
        return x==0 ? 64 : __builtin_ctzll(u64(x));
    else if(is_same<Int,i32>::value)
        return x==0 ? 32 : __builtin_clz(u32(x));
    ASSERT(false);
}

/*constexpr*/ Int BIT_COUNT_ONES(Int x) {
    if(is_same<Int,i64>::value)
        return __builtin_popcountll(u64(x));
    else if(is_same<Int,i32>::value)
        return __builtin_popcount(u32(x));
    ASSERT(false);
}

// 1の個数が奇数なら1, 偶数なら0を返す
/*constexpr*/ Int BIT_PARITY(Int x) {
    if(is_same<Int,i64>::value)
        return __builtin_parityll(u64(x));
    else if(is_same<Int,i32>::value)
        return __builtin_parity(u32(x));
    ASSERT(false);
}

// 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
//
// ```
// Int x = BIT_I_1(3);
// do {
//     // ...
// } while(BIT_NEXT_SET_SIZED(x, 10));
// ```
/*constexpr*/ bool BIT_NEXT_SET_SIZED(Int& x, Int n) {
    if(x == 0) return false;
    Int t = (x|(x-1)) + 1;
    x = t | ((~t&(t-1)) >> (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
//
// ```
// Int y = 0b10101;
// Int x = 0;
// do {
//     // ...
// } while(BIT_NEXT_SUBSET(x, y));
// ```
constexpr bool BIT_NEXT_SUBSET(Int& x, Int 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
//
// ```
// Int y = 0b10101;
// Int x = y;
// do {
//     // ...
// } while(BIT_PREV_SUBSET(x, y));
// ```
constexpr bool BIT_PREV_SUBSET(Int& x, Int 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
//
// ```
// Int y = 0b00010101;
// Int x = y;
// do {
//     // ...
// } while(BIT_NEXT_SUPERSET(x, 8, y));
// ```
constexpr bool BIT_NEXT_SUPERSET(Int& x, Int n, Int y) {
    x = (x+1) | y;
    return x < BIT_I(n);
}
// }}}

// lo:OK, hi:NG
template<class Pred>
/*constexpr*/ Int bisect_integer(Int lo, Int hi, Pred pred) {
    ASSERT(lo < hi);

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

template<class Pred>
/*constexpr*/ Real bisect_real(Real lo, Real hi, Pred pred, Real eps=EPS) {
    ASSERT_LOCAL(!GT_EPS(lo,hi,eps));
    if(lo > hi) swap(lo, hi);

    while(!EQ_EPS(lo,hi,eps)) {
        Real mid = (lo+hi) / 2;
        if(pred(mid))
            lo = mid;
        else
            hi = mid;
    }
    return lo;
}

template<class Monoid>
/*constexpr*/ Monoid fastpow(const Monoid& x, Int e, const Monoid& unity) {
    ASSERT(e >= 0);

    Monoid res = unity;
    Monoid cur = x;
    while(e > 0) {
        if(e & 1)
            res *= cur;
        cur *= cur;
        e >>= 1;
    }
    return res;
}

/*constexpr*/ Int ipow(Int x, Int e) {
    return fastpow<Int>(x,e,1);
}

/*constexpr*/ Int sqrt_floor(Int x) {
    ASSERT(x >= 0);

    Int lo = 0;
    Int hi = MIN(x/2+2, SQRT_MAX<Int>()+1);
    return bisect_integer(lo, hi, [x](Int r) { return r*r <= x; });
}

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

/*constexpr*/ Int log2_ceil(Int x) {
    ASSERT(x > 0);
    if(is_same<Int,i64>::value)
        return 64 - BIT_COUNT_LEADING_ZEROS(x-1);
    else if(is_same<Int,i32>::value)
        return 32 - BIT_COUNT_LEADING_ZEROS(x-1);
    ASSERT(false);
}

/*constexpr*/ Int log2_floor(Int x) {
    ASSERT(x > 0);
    if(is_same<Int,i64>::value)
        return 63 - BIT_COUNT_LEADING_ZEROS(x);
    else if(is_same<Int,i32>::value)
        return 31 - BIT_COUNT_LEADING_ZEROS(x);
    ASSERT(false);
}

// x > 0
/*constexpr*/ Int pow2_ceil(Int x) {
    return BIT_I(log2_ceil(x));
}

// x > 0
/*constexpr*/ Int pow2_floor(Int x) {
    return BIT_I(log2_floor(x));
}

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

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

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

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

/*constexpr*/ Int align_ceil(Int x, Int align) {
    ASSERT(align > 0);
    return div_ceil(x,align) * align;
}

/*constexpr*/ Int align_floor(Int x, Int align) {
    ASSERT(align > 0);
    return div_floor(x,align) * align;
}

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

template<class InputIt>
auto SUM(InputIt first, InputIt last) {
    return FOLD1(first, last, plus<>{});
}

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

template<class C>
void SORT_UNIQ(C& c) {
    ALL(sort, c);
    UNIQ(c);
}

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

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

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

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

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

// container {{{
template<class T> struct is_container : false_type {};
template<class T, size_t N> struct is_container<array<T,N>> : true_type {};
template<class... Args> struct is_container<vector<Args...>> : true_type {};
template<class... Args> struct is_container<deque<Args...>> : true_type {};
template<class... Args> struct is_container<list<Args...>> : true_type {};
template<class... Args> struct is_container<forward_list<Args...>> : true_type {};
template<class... Args> struct is_container<set<Args...>> : true_type {};
template<class... Args> struct is_container<multiset<Args...>> : true_type {};
template<class... Args> struct is_container<unordered_set<Args...>> : true_type {};
template<class... Args> struct is_container<unordered_multiset<Args...>> : true_type {};
template<class... Args> struct is_container<map<Args...>> : true_type {};
template<class... Args> struct is_container<multimap<Args...>> : true_type {};
template<class... Args> struct is_container<unordered_map<Args...>> : true_type {};
template<class... Args> struct is_container<unordered_multimap<Args...>> : true_type {};

template<class T, class Enable=void>
struct ProconHash {
    size_t operator()(const T& x) const noexcept {
        return hash<T>{}(x);
    }
};

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

size_t procon_hash_combine(size_t h1, size_t h2) noexcept {
    constexpr size_t M = UINT64_C(0xc6a4a7935bd1e995);
    constexpr int    R = 47;

    h2 *= M;
    h2 ^= h2 >> R;
    h2 *= M;

    h1 ^= h2;
    h1 *= M;

    h1 += 0xe6546b64;

    return h1;
}

template<class T1, class T2>
struct ProconHash<pair<T1,T2>> {
    size_t operator()(const pair<T1,T2>& p) const noexcept {
        size_t h1 = procon_hash_value(p.first);
        size_t h2 = procon_hash_value(p.second);
        return procon_hash_combine(h1, h2);
    }
};

template<class... TS>
struct ProconHash<tuple<TS...>> {
    size_t operator()(const tuple<TS...>& t) const noexcept {
        size_t h = 0;
        tuple_enumerate(t, [&h](Int, const auto& e) {
            h = procon_hash_combine(h, procon_hash_value(e));
        });
        return h;
    }
};

template<class C>
struct ProconHash<C,enable_if_t<is_container<C>::value>> {
    size_t operator()(const C& c) const noexcept {
        size_t h = 0;
        for(const auto& e : c)
            h = procon_hash_combine(h, procon_hash_value(e));
        return h;
    }
};

template<class T, class Hash=ProconHash<T>, class Eq=equal_to<T>>
using HashSet = unordered_set<T,Hash,Eq>;
template<class K, class V, class Hash=ProconHash<K>, class Eq=equal_to<K>>
using HashMap = unordered_map<K,V,Hash,Eq>;
template<class T, class Hash=ProconHash<T>, class Eq=equal_to<T>>
using HashMultiset = unordered_multiset<T,Hash,Eq>;
template<class K, class V, class Hash=ProconHash<K>, class Eq=equal_to<K>>
using HashMultimap = unordered_multimap<K,V,Hash,Eq>;

template<class T>
auto vec_make(Int n, T x) {
    return vector<T>(n, x);
}

template<class T, class... Args, SFINAE(sizeof...(Args) >= 2)>
auto vec_make(Int n, Args... args) {
    auto inner = vec_make<T>(args...);
    return vector<decltype(inner)>(n, inner);
}

template<class T>
auto vec_reserve(Int cap) {
    vector<T> res;
    res.reserve(cap);
    return res;
}

template<class T=Int>
auto vec_iota(Int n, T init={}) {
    vector<Int> res(n);
    ALL(iota, res, init);
    return res;
}

template<class ForwardIt, class BinaryOp>
auto vec_scan(ForwardIt first, ForwardIt last,
              typename iterator_traits<ForwardIt>::value_type init,
              BinaryOp op)
{
    using T = typename iterator_traits<ForwardIt>::value_type;
    auto res = vec_reserve<T>(distance(first,last)+1);
    res.emplace_back(init);
    for(; first != last; ++first) {
        init = op(move(init), *first);
        res.emplace_back(init);
    }
    return res;
}

template<class ForwardIt>
auto vec_cum(ForwardIt first, ForwardIt last) {
    using T = typename iterator_traits<ForwardIt>::value_type;
    return vec_scan(first, last, T{}, plus<>{});
}

template<class T, class Comp, class Cont=vector<T>>
auto priority_queue_make(const Comp& comp, Cont&& cont={}) {
    return priority_queue<T,remove_reference_t<Cont>,Comp>(comp, forward<Cont>(cont));
}

template<class T, class Comp>
auto priority_queue_reserve(const Comp& comp, Int cap) {
    return priority_queue<T,vector<T>,Comp>(comp, vec_reserve<T>(cap));
}

template<class T, size_t N, size_t... NS>
struct ArrayType {
    using type = array<class ArrayType<T,NS...>::type,N>;
};

template<class T, size_t N>
struct ArrayType<T,N> {
    using type = array<T,N>;
};

template<class T, size_t... NS>
using Array = typename ArrayType<T,NS...>::type;

template<class T, size_t N>
T& array_at(Array<T,N>& ary, Int i) {
    return ary[i];
}

template<class T, size_t N, size_t... NS, class... Args>
T& array_at(Array<T,N,NS...>& ary, Int i, Args... args) {
    return array_at<T,NS...>(ary[i], args...);
}

template<class T, size_t N>
const T& array_at(const Array<T,N>& ary, Int i) {
    return ary[i];
}

template<class T, size_t N, size_t... NS, class... Args>
const T& array_at(const Array<T,N,NS...>& ary, Int i, Args... args) {
    return array_at<T,NS...>(ary[i], args...);
}

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

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

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

// fixpoint {{{
template<class F>
class FixPoint {
public:
    explicit constexpr FixPoint(F&& f) : f_(forward<F>(f)) {}

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

private:
    F f_;
};

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

template<class F, size_t... NS>
class FixPointMemo {
public:
    explicit FixPointMemo(F&& f) : f_(forward<F>(f)) {}

    template<class... Args>
    decltype(auto) operator()(Args... args) const {
        using R = decltype(f_(*this,args...));
        static Array<bool,NS...> done {};
        static Array<R,NS...>    memo;

        if(!array_at<bool,NS...>(done,args...)) {
            array_at<R,NS...>(memo,args...) = f_(*this,args...);
            array_at<bool,NS...>(done,args...) = true;
        }
        return array_at<R,NS...>(memo,args...);
    }

private:
    F f_;
};

template<size_t... NS, class F>
decltype(auto) FIXMEMO(F&& f) {
    return FixPointMemo<F,NS...>(forward<F>(f));
}
// }}}

// math {{{
/*constexpr*/ Int GCD(Int a, Int b) noexcept {
    /*constexpr*/ auto f_gcd = FIX([](auto&& self, Int aa, Int bb) -> Int {
        if(bb == 0) return aa;
        return self(bb, aa%bb);
    });
    return f_gcd(ABS(a), ABS(b));
}

/*constexpr*/ Int LCM(Int a, Int b) noexcept {
    ASSERT(a != 0 && b != 0);
    /*constexpr*/ auto f_gcd = FIX([](auto&& self, Int aa, Int bb) -> Int {
        if(bb == 0) return aa;
        return self(bb, aa%bb);
    });
    a = ABS(a);
    b = ABS(b);
    return a / f_gcd(a,b) * b;
}

/*constexpr*/ tuple<Int,Int,Int> EXTGCD(Int a, Int b) noexcept {
    /*constexpr*/ auto impl = FIX([](auto&& self, Int aa, Int bb, Int& x, Int& y) -> Int {
        if(bb == 0) {
            x = 1; y = 0;
            return aa;
        }
        Int g = self(bb, aa%bb, y, x);
        y -= (aa/bb)*x;
        return g;
    });
    Int x{},y{};
    Int g = impl(ABS(a), ABS(b), x, y);
    x *= SGN(a);
    y *= SGN(b);
    return make_tuple(g, x, y);
}
// }}}

// string {{{
char chr_digit(Int n)  { return char('0'+n); }
Int  ord_digit(char c) { return c-'0'; }
char chr_lower(Int n)  { return char('a'+n); }
Int  ord_lower(char c) { return c-'a'; }
char chr_upper(Int n)  { return char('A'+n); }
Int  ord_upper(char c) { return c-'A'; }

auto str_reserve(Int cap) {
    string res;
    res.reserve(cap);
    return res;
}
// }}}

// input {{{
template<class T>
struct Integral1 {
    static_assert(is_integral<T>::value && !is_same<T,bool>::value, "");
};
using Int1 = Integral1<Int>;

template<class T, class Enable=void>
struct Scan {
    using R = T;
    static R scan(istream& in) {
        R res;
        in >> res;
        return res;
    }
};

template<class T>
struct Scan<Integral1<T>> {
    using R = T;
    static R scan(istream& in) {
        return Scan<R>::scan(in) - 1;
    }
};

template<class T1, class T2>
struct Scan<pair<T1,T2>> {
    using R1 = typename Scan<T1>::R;
    using R2 = typename Scan<T2>::R;
    using R = pair<R1,R2>;
    static R scan(istream& in) {
        R1 x = Scan<T1>::scan(in);
        R2 y = Scan<T2>::scan(in);
        return {x,y};
    }
};

template<class T>
auto tuple_scan_impl(istream& in) {
    return make_tuple(Scan<T>::scan(in));
}

template<class T, class... TS, SFINAE(sizeof...(TS) > 0)>
auto tuple_scan_impl(istream& in) {
    auto head = make_tuple(Scan<T>::scan(in));
    return tuple_cat(head, tuple_scan_impl<TS...>(in));
}

template<class... TS>
struct Scan<tuple<TS...>> {
    using R = decltype(tuple_scan_impl<TS...>(cin));
    static R scan(istream& in) {
        return tuple_scan_impl<TS...>(in);
    }
};

template<class T=Int>
auto RD() {
    return Scan<T>::scan(cin);
}

template<class T=Int>
auto RD1() {
    return RD<Integral1<T>>();
}

template<class T=Int>
auto RD_VEC(Int n) {
    auto res = vec_reserve<typename Scan<T>::R>(n);
    LOOP(n) {
        res.emplace_back(RD<T>());
    }
    return res;
}

template<class T=Int>
auto RD1_VEC(Int n) {
    return RD_VEC<Integral1<T>>(n);
}

template<class T=Int>
auto RD_VEC2(Int h, Int w) {
    auto res = vec_reserve<vector<typename Scan<T>::R>>(h);
    LOOP(h) {
        res.emplace_back(RD_VEC<T>(w));
    }
    return res;
}

template<class T=Int>
auto RD1_VEC2(Int h, Int w) {
    return RD_VEC2<Integral1<T>>(h, w);
}
// }}}

// output {{{
template<class T, class Enable=void>
struct Fmt {
    static void fmt(ostream& out, const T& x) { out << x; }
};

template<class T>
void fmt_write(ostream& out, const T& x) {
    Fmt<T>::fmt(out, x);
}

template<class T>
string FMT_STR(const T& x) {
    ostringstream out;
    fmt_write(out, x);
    return out.str();
}

template<class... TS>
struct Fmt<tuple<TS...>> {
    static void fmt(ostream& out, const tuple<TS...>& t) {
        tuple_enumerate(t, [&out](Int i, const auto& e) {
            if(i != 0) out << ' ';
            fmt_write(out, e);
        });
    }
};

template<class T1, class T2>
struct Fmt<pair<T1,T2>> {
    static void fmt(ostream& out, const pair<T1,T2>& p) {
        return fmt_write(out, make_tuple(p.first,p.second));
    }
};

template<class C>
struct Fmt<C,enable_if_t<is_container<C>::value>> {
    static void fmt(ostream& out, const C& c) {
        for(auto it = begin(c); it != end(c); ++it) {
            if(it != begin(c)) out << ' ';
            fmt_write(out, *it);
        }
    }
};

void PRINT() {}

template<class T, class... TS>
void PRINT(const T& x, const TS&... args) {
    fmt_write(cout, x);
    if(sizeof...(args) > 0) {
        cout << ' ';
        PRINT(args...);
    }
}

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

// debug {{{
template<class T, class Enable=void>
struct Dbg {
    static void dbg(ostream& out, const T& x) { out << x; }
};

template<class T>
void dbg_write(ostream& out, const T& x) {
    Dbg<T>::dbg(out, x);
}

template<class T>
string DBG_STR(const T& x) {
    ostringstream out;
    dbg_write(out, x);
    return out.str();
}

template<>
struct Dbg<Int> {
    static void dbg(ostream& out, Int x) {
        if(x == INF)
            out << "INF";
        else if(x == -INF)
            out << "-INF";
        else
            out << x;
    }
};

template<>
struct Dbg<Real> {
    static void dbg(ostream& out, Real x) {
        if(EQ_EXACT(x, FINF))
            out << "FINF";
        else if(EQ_EXACT(x, -FINF))
            out << "-FINF";
        else
            out << x;
    }
};

template<class T, size_t N>
struct Dbg<T[N]> {
    static void dbg(ostream& out, const T (&ary)[N]) {
        out << "[";
        REP(i, N) {
            if(i != 0) out << ",";
            dbg_write(out, ary[i]);
        }
        out << "]";
    }
};

template<size_t N>
struct Dbg<char[N]> {
    static void dbg(ostream& out, const char (&s)[N]) {
        out << s;
    }
};

template<class... TS>
struct Dbg<tuple<TS...>> {
    static void dbg(ostream& out, const tuple<TS...>& t) {
        out << "(";
        tuple_enumerate(t, [&out](Int i, const auto& e) {
            if(i != 0) out << ",";
            dbg_write(out, e);
        });
        out << ")";
    }
};

template<class T1, class T2>
struct Dbg<pair<T1,T2>> {
    static void dbg(ostream& out, const pair<T1,T2>& p) {
        return dbg_write(out, make_tuple(p.first,p.second));
    }
};

template<class C>
struct Dbg<C,enable_if_t<is_container<C>::value>> {
    static void dbg(ostream& out, const C& c) {
        out << "[";
        for(auto it = begin(c); it != end(c); ++it) {
            if(it != begin(c)) out << ",";
            dbg_write(out, *it);
        }
        out << "]";
    }
};

template<class T, class C>
struct Dbg<stack<T,C>> {
    static void dbg(ostream& out, stack<T,C> stk) {
        out << "[";
        while(!stk.empty()) {
            dbg_write(out,stk.top()); stk.pop();
            if(!stk.empty()) out << ",";
        }
        out << "]";
    }
};

template<class T, class C>
struct Dbg<queue<T,C>> {
    static void dbg(ostream& out, queue<T,C> que) {
        out << "[";
        while(!que.empty()) {
            dbg_write(out,que.front()); que.pop();
            if(!que.empty()) out << ",";
        }
        out << "]";
    }
};

template<class T, class C, class Comp>
struct Dbg<priority_queue<T,C,Comp>> {
    static void dbg(ostream& out, priority_queue<T,C,Comp> que) {
        out << "[";
        while(!que.empty()) {
            dbg_write(out,que.top()); que.pop();
            if(!que.empty()) out << ",";
        }
        out << "]";
    }
};

template<class T>
void DBG_IMPL(Int line, const char* expr, const T& value) {
    cerr << "[L " << line << "]: ";
    cerr << expr << " = ";
    dbg_write(cerr, value);
    cerr << "\n";
}

void DBG_IMPL_HELPER() {}

template<class T, class... TS>
void DBG_IMPL_HELPER(const T& x, const TS&... args) {
    dbg_write(cerr, x);
    if(sizeof...(args) > 0) {
        cerr << ",";
        DBG_IMPL_HELPER(args...);
    }
}

template<class... TS>
void DBG_IMPL(Int line, const char* expr, const TS&... value) {
    cerr << "[L " << line << "]: ";
    cerr << "(" << expr << ") = (";
    DBG_IMPL_HELPER(value...);
    cerr << ")\n";
}

template<size_t N, class T, SFINAE(rank<T>::value == 0)>
void DBG_DP_IMPL_HELPER(ostream& out, const T& x, const array<Int,N>&, const array<Int,N>&) {
    dbg_write(out, x);
}

template<size_t N, class T, SFINAE(rank<T>::value > 0)>
void DBG_DP_IMPL_HELPER(ostream& out, const T& x, const array<Int,N>& sizes, const array<Int,N>& offs) {
    Int k   = N - rank<T>::value;
    Int off = offs[k];
    Int siz = sizes[k];
    if(siz == 0) siz = extent<T>::value - off;

    out << "[";
    FOR(i, off, off+siz) {
        if(i != off) out << ",";
        DBG_DP_IMPL_HELPER(out, x[i], sizes, offs);
    }
    out << "]";
}

template<class T, SFINAE(rank<T>::value > 0)>
void DBG_DP_IMPL(Int line, const char* expr, const T& dp,
                 const array<Int,rank<T>::value>& sizes={},
                 const array<Int,rank<T>::value>& offs={})
{
    cerr << "[L " << line << "]: ";
    cerr << expr << " = ";
    DBG_DP_IMPL_HELPER<rank<T>::value>(cerr, dp, sizes, offs);
    cerr << "\n";
}

template<class T>
void DBG_GRID_IMPL(Int line, const char* expr, const vector<T>& grid) {
    cerr << "[L " << line << "]: ";
    cerr << expr << ":\n";
    for(const auto& row : grid) {
        dbg_write(cerr, row);
        cerr << "\n";
    }
    cerr << "\n";
}

#ifdef PROCON_LOCAL
    #define DBG(args...) DBG_IMPL(__LINE__, CPP_STR_I(args), args)
    #define DBG_DP(args...) DBG_DP_IMPL(__LINE__, CPP_STR_I(args), args)
    #define DBG_GRID(args...) DBG_GRID_IMPL(__LINE__, CPP_STR_I(args), args)
#else
    #define DBG(args...)
    #define DBG_DP(args...)
    #define DBG_GRID(args...)
#endif
// }}}

// modint {{{
template<class Mod>
class ModIntT {
private:
    Int v_;  // [0,Mod::value)

    static Int mod() { return Mod::value; }

    static Int normalize(Int x) {
        Int res = x % mod();
        if(res < 0) res += mod();
        return res;
    }

public:
    ModIntT() : v_(0) {}
    ModIntT(Int v) : v_(normalize(v)) {}

    explicit operator Int() const { return v_; }

    ModIntT operator-() const { return ModIntT(-v_); }

    ModIntT& operator+=(ModIntT rhs) {
        v_ = normalize(v_ + rhs.v_);
        return *this;
    }
    ModIntT& operator-=(ModIntT rhs) {
        v_ = normalize(v_ - rhs.v_);
        return *this;
    }
    ModIntT& operator*=(ModIntT rhs) {
        v_ = normalize(v_ * rhs.v_);
        return *this;
    }

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

    ModIntT pow(Int e) const {
        return fastpow(*this, e, ModIntT(1));
    }

    ModIntT inv() const {
        Int g,x; tie(g,x,ignore) = EXTGCD(v_, mod());
        ASSERT(g == 1);
        return ModIntT(x);
    }

    friend ModIntT operator+(ModIntT lhs, ModIntT rhs) { return ModIntT(lhs) += rhs; }
    friend ModIntT operator+(ModIntT lhs, Int rhs)     { return ModIntT(lhs) += rhs; }
    friend ModIntT operator+(Int lhs, ModIntT rhs)     { return ModIntT(rhs) += lhs; }
    friend ModIntT operator-(ModIntT lhs, ModIntT rhs) { return ModIntT(lhs) -= rhs; }
    friend ModIntT operator-(ModIntT lhs, Int rhs)     { return ModIntT(lhs) -= rhs; }
    friend ModIntT operator-(Int lhs, ModIntT rhs)     { return ModIntT(rhs) -= lhs; }
    friend ModIntT operator*(ModIntT lhs, ModIntT rhs) { return ModIntT(lhs) *= rhs; }
    friend ModIntT operator*(ModIntT lhs, Int rhs)     { return ModIntT(lhs) *= rhs; }
    friend ModIntT operator*(Int lhs, ModIntT rhs)     { return ModIntT(rhs) *= lhs; }

    friend bool operator==(ModIntT lhs, ModIntT rhs) { return Int(lhs) == Int(rhs); }
    friend bool operator==(ModIntT lhs, Int rhs)     { return lhs == ModIntT(rhs); }
    friend bool operator==(Int lhs, ModIntT rhs)     { return ModIntT(lhs) == rhs; }
    friend bool operator!=(ModIntT lhs, ModIntT rhs) { return !(lhs == rhs); }
    friend bool operator!=(ModIntT lhs, Int rhs)     { return !(lhs == rhs); }
    friend bool operator!=(Int lhs, ModIntT rhs)     { return !(lhs == rhs); }
};

template<class Mod>
struct ProconHash<ModIntT<Mod>> {
    size_t operator()(ModIntT<Mod> x) const noexcept {
        return procon_hash_value(Int(x));
    }
};

template<class Mod>
struct Scan<ModIntT<Mod>> {
    using R = ModIntT<Mod>;
    static R scan(istream& in) {
        Int v = Scan<Int>::scan(in);
        return ModIntT<Mod>(v);
    }
};

template<class Mod>
struct Fmt<ModIntT<Mod>> {
    static void fmt(ostream& out, ModIntT<Mod> x) {
        fmt_write(out, Int(x));
    }
};

template<class Mod>
struct Dbg<ModIntT<Mod>> {
    static void dbg(ostream& out, ModIntT<Mod> x) {
        dbg_write(out, Int(x));
    }
};

template<Int M>
using ModIntC = ModIntT<integral_constant<Int,M>>;

using ModInt = ModIntC<MOD>;
// }}}

// rng {{{
// http://prng.di.unimi.it/xoroshiro128plus.c
struct ProconUrbg {
    using result_type = u64;
    static constexpr result_type min() { return numeric_limits<result_type>::min(); }
    static constexpr result_type max() { return numeric_limits<result_type>::max(); }

    ProconUrbg(u64 s0, u64 s1) : state_{s0,s1} {}

    result_type operator()() {
        u64 s0 = state_[0];
        u64 s1 = state_[1];
        u64 res = s0 + s1;

        s1 ^= s0;
        state_[0] = ((s0<<24)|(s0>>40)) ^ s1 ^ (s1<<16);
        state_[1] = (s1<<37)|(s1>>27);

        return res;
    }

private:
    u64 state_[2];
};

ProconUrbg& PROCON_URBG() {
    static u64 s0 = u64(chrono::system_clock::now().time_since_epoch().count());
    static u64 s1 = u64(&s0);
    static ProconUrbg urbg(s0, s1);
    return urbg;
}
// }}}

// init {{{
struct ProconInit {
    ProconInit() {
        cin.tie(nullptr);
        ios::sync_with_stdio(false);
        cin.exceptions(ios::failbit | ios::badbit);
        cout << fixed << setprecision(COUT_PREC);
#ifdef PROCON_LOCAL
        cerr << fixed << setprecision(2);
#endif
        if(COUT_AUTOFLUSH)
            cout << unitbuf;
    }
} PROCON_INIT;
// }}}

// }}}

// graph {{{

auto udgraph_list(Int n, const vector<pair<Int,Int>>& es) {
    vector<vector<Int>> g(n);
    for(const auto& e : es) {
        Int a,b; tie(a,b) = e;
        g[a].emplace_back(b);
        g[b].emplace_back(a);
    }
    return g;
}

auto digraph_list(Int n, const vector<pair<Int,Int>>& es) {
    vector<vector<Int>> g(n);
    for(const auto& e : es) {
        Int a,b; tie(a,b) = e;
        g[a].emplace_back(b);
    }
    return g;
}

auto udgraph_matrix(Int n, const vector<pair<Int,Int>>& es) {
    vector<vector<Int>> g(n, vector<Int>(n,INF));
    REP(i, n) { g[i][i] = 0; }
    for(const auto& e : es) {
        Int a,b; tie(a,b) = e;
        g[a][b] = g[b][a] = 1;
    }
    return g;
}

auto digraph_matrix(Int n, const vector<pair<Int,Int>>& es) {
    vector<vector<Int>> g(n, vector<Int>(n,INF));
    REP(i, n) { g[i][i] = 0; }
    for(const auto& e : es) {
        Int a,b; tie(a,b) = e;
        g[a][b] = 1;
    }
    return g;
}

template<class T>
auto wudgraph_list(Int n, const vector<tuple<Int,Int,T>>& es) {
    vector<vector<pair<Int,T>>> g(n);
    for(const auto& e : es) {
        Int a,b; T c; tie(a,b,c) = e;
        g[a].emplace_back(b, c);
        g[b].emplace_back(a, c);
    }
    return g;
}

template<class T>
auto wdigraph_list(Int n, const vector<tuple<Int,Int,T>>& es) {
    vector<vector<pair<Int,T>>> g(n);
    for(const auto& e : es) {
        Int a,b; T c; tie(a,b,c) = e;
        g[a].emplace_back(b, c);
    }
    return g;
}

template<class T>
auto wudgraph_matrix(Int n, const vector<tuple<Int,Int,T>>& es) {
    vector<vector<T>> g(n, vector<T>(n,PROCON_INF<T>()));;
    REP(i, n) { g[i][i] = T{}; }
    for(const auto& e : es) {
        Int a,b; T c; tie(a,b,c) = e;
        g[a][b] = g[b][a] = c;
    }
    return g;
}

template<class T>
auto wdigraph_matrix(Int n, const vector<tuple<Int,Int,T>>& es) {
    vector<vector<T>> g(n, vector<T>(n,PROCON_INF<T>()));;
    REP(i, n) { g[i][i] = T{}; }
    for(const auto& e : es) {
        Int a,b; T c; tie(a,b,c) = e;
        g[a][b] = c;
    }
    return g;
}

// 単純無向グラフが木かどうか判定する
//
// g: 隣接リスト表現(頂点数 n > 0)
bool graph_is_tree(const vector<vector<Int>>& g) {
    Int n = SIZE(g);
    ASSERT(n > 0);

    Int edge_cnt = 0;
    vector<bool> visited(n, false);
    auto dfs = FIX([&g,&edge_cnt,&visited](auto&& self, Int v) -> void {
        visited[v] = true;
        for(Int to : g[v]) {
            if(visited[to]) continue;
            ++edge_cnt;
            self(to);
        }
    });
    dfs(0);

    bool connected = ALL(all_of, visited, [](bool b) { return b; });

    return edge_cnt == n-1 && connected;
}

// BFSで重みなしグラフ上の単一始点最短経路を求める
//
// (ds,ps) を返す
// ds[i]: start から点 i への最短距離(到達不能な点は INF)
// ps[i]: 最短経路木における点 i の親(start および到達不能な点は -1)
tuple<vector<Int>,vector<Int>> graph_bfs(const vector<vector<Int>>& g, Int start) {
    Int n = SIZE(g);
    vector<Int> ds(n, INF);
    vector<Int> ps(n, -1);

    queue<Int> que;
    que.emplace(start);
    ds[start] = 0;

    while(!que.empty()) {
        Int v = POP(que);

        for(Int to : g[v]) {
            if(ds[to] != INF) continue;
            que.emplace(to);
            ds[to] = ds[v] + 1;
            ps[to] = v;
        }
    }

    return make_tuple(ds, ps);
}

// ダイクストラ法
//
// (ds,ps) を返す
// ds[i]: start から点 i への最短距離(到達不能な点は PROCON_INF<T>())
// ps[i]: 最短経路木における点 i の親(start および到達不能な点は -1)
template<typename T>
tuple<vector<T>,vector<Int>> graph_dijkstra(const vector<vector<pair<Int,T>>>& g, Int start) {
    Int n = SIZE(g);
    vector<T> ds(n, PROCON_INF<T>());
    vector<Int> ps(n, -1);

    auto que = priority_queue_make<pair<T,Int>>(greater<>{});

    ds[start] = T{};
    que.emplace(T{}, start);

    Int n_remain = n;
    while(!que.empty()) {
        T d; Int v; tie(d,v) = POP(que);
        if(ds[v] < d) continue;

        if(--n_remain == 0) break;

        for(const auto& p : g[v]) {
            Int to; T cost; tie(to,cost) = p;

            T d_new = d + cost;
            if(chmin(ds[to], d_new)) {
                ps[to] = v;
                que.emplace(d_new, to);
            }
        }
    }

    return make_tuple(ds, ps);
}

// 辺のコストが小さい非負整数の場合の最良優先探索(01-BFS の一般化)
// 全ての辺のコストは [0,k] であること
//
// (ds,ps) を返す
// ds[i]: start から点 i への最短距離(到達不能な点は INF)
// ps[i]: 最短経路木における点 i の親(start および到達不能な点は -1)
tuple<vector<Int>,vector<Int>> graph_k_bfs(const vector<vector<pair<Int,Int>>>& g, Int k, Int start) {
    Int n = SIZE(g);
    vector<Int> ds(n, INF);
    vector<Int> ps(n, -1);

    vector<queue<Int>> ques(k+1);
    auto enqueue = [&ques](Int to, Int cost) {
        ques[cost].emplace(to);
    };
    auto dequeue = [&ques]() -> Int {
        for(auto& que : ques)
            if(!que.empty())
                return POP(que);
        return -1;
    };

    enqueue(start, 0);
    ds[start] = 0;

    Int v;
    while((v = dequeue()) != -1) {
        for(const auto& p : g[v]) {
            Int to,cost; tie(to,cost) = p;

            Int d_new = ds[v] + cost;
            if(chmin(ds[to], d_new)) {
                ps[to] = v;
                enqueue(to, cost);
            }
        }
    }

    return make_tuple(ds, ps);
}

// ベルマンフォード法
//
// 負閉路が存在する場合、最短距離が負の無限大になる点が生じる。
// そのような点を全て検出するため、2*n 回ループしている
// (一般的な実装の倍の回数。ただし更新がなくなったら打ち切る)
//
// (ds,ps) を返す
// ds[i]: start から点 i への最短距離(到達不能なら INF, 負の無限大なら -INF)
// ps[i]: 最短経路木における点 i の親(start および到達不能な点は -1)
template<typename T>
tuple<vector<T>,vector<Int>> graph_bellman(const vector<vector<pair<Int,T>>>& g, Int start) {
    Int n = SIZE(g);
    vector<T> ds(n, PROCON_INF<T>());
    vector<Int> ps(n, -1);

    ds[start] = T{};

    REP(i, 2*n) {
        bool update = false;
        REP(from, n) {
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
            if(ds[from] == PROCON_INF<T>()) continue;
            for(const auto& p : g[from]) {
                Int to; T cost; tie(to,cost) = p;
                T d_new = ds[from]==-PROCON_INF<T>() ? -PROCON_INF<T>() : ds[from]+cost;
                if(d_new < ds[to]) {
                    update = true;
                    ds[to] = i >= n-1 ? -PROCON_INF<T>() : d_new;
                    ps[to] = from;
                }
            }
#pragma GCC diagnostic pop
        }
        if(!update) break;
    }

    return make_tuple(ds, ps);
}

// SPFA (Shortest Path Faster Algorithm)
//
// 理論上はベルマンフォードより速いはずだが、実際はそうでもなさげ
// 最短距離が負の無限大になる点を全て検出するため 2*n 回ループしている
// (一般的な実装の倍の回数。ただし更新がなくなったら打ち切る)
//
// (ds,ps) を返す
// ds[i]: start から点 i への最短距離(到達不能なら INF, 負の無限大なら -INF)
// ps[i]: 最短経路木における点 i の親(start および到達不能な点は -1)
template<typename T>
tuple<vector<T>,vector<Int>> graph_spfa(const vector<vector<pair<Int,T>>>& g, Int start) {
    Int n = SIZE(g);
    vector<T> ds(n, PROCON_INF<T>());
    vector<Int> ps(n, -1);

    queue<Int> que;
    vector<bool> in_que(n, false);
    const auto enqueue = [&que,&in_que](Int v) { que.emplace(v); in_que[v] = true; };
    const auto dequeue = [&que,&in_que]() { Int v = POP(que); in_que[v] = false; return v; };

    ds[start] = T{};
    enqueue(start);

    REP(i, 2*n) {
        REP(_, que.size()) {
            Int from = dequeue();
            for(const auto& p : g[from]) {
                Int to; T cost; tie(to,cost) = p;
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
                T d_new = ds[from]==-PROCON_INF<T>() ? -PROCON_INF<T>() : ds[from]+cost;
                if(d_new < ds[to]) {
                    ds[to] = i >= n-1 ? -PROCON_INF<T>() : d_new;
                    ps[to] = from;
                    if(!in_que[to]) enqueue(to);
                }
#pragma GCC diagnostic pop
            }
        }
        if(que.empty()) break;
    }

    return make_tuple(ds, ps);
}

// ワーシャルフロイド法
//
// g は隣接行列 (g[from][to]) で、from == to の場合 0, from != to で辺
// がない場合 INF
//
// g は全点対間最短距離で上書きされる
// (ok,nex) を返す
// ok: 負閉路が存在しない場合に限り true
// nex[i][j]: i から j へ最短経路で行くとき、次に辿るべき点(到達不能なら -1)
template<typename T>
tuple<bool,vector<vector<Int>>> graph_floyd(vector<vector<T>>& g) {
    Int n = SIZE(g);
    vector<vector<Int>> nex(n, vector<Int>(n,-1));

#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
    REP(i, n) REP(j, n) {
        if(g[i][j] != PROCON_INF<T>())
            nex[i][j] = j;
    }

    REP(k, n) {
        REP(i, n) {
            if(g[i][k] == PROCON_INF<T>()) continue;
            REP(j, n) {
                if(g[k][j] == PROCON_INF<T>()) continue;
                if(chmin(g[i][j], g[i][k] + g[k][j])) {
                    nex[i][j] = nex[i][k];
                }
                if(i == j && g[i][j] < 0) return make_tuple(false, nex);
            }
        }
    }
#pragma GCC diagnostic pop

    return make_tuple(true, nex);
}

// TODO: 重みあり/なし両対応
// トポロジカルソート
// queue を MinHeap に変えると辞書順最小のものが求まる
//
// (ok,res) を返す
// ok: DAGであったかどうか
// res: 結果
tuple<bool,vector<Int>> graph_tsort(const vector<vector<Int>>& g) {
    Int n = SIZE(g);
    vector<Int> res;
    res.reserve(n);

    vector<Int> deg_in(n, 0);
    for(const auto& tos : g)
        for(auto to : tos)
            ++deg_in[to];

    queue<Int> que;
    REP(v, n) {
        if(deg_in[v] == 0)
            que.emplace(v);
    }

    while(!que.empty()) {
        Int v = POP(que);

        res.emplace_back(v);

        for(auto to : g[v]) {
            if(--deg_in[to] > 0) continue;
            que.emplace(to);
        }
    }

    bool ok = SIZE(res) == n;
    return make_tuple(ok, res);
}

// TODO: 重みあり/なし両対応
// (関節点リスト,橋リスト) を返す
tuple<vector<Int>,vector<pair<Int,Int>>> graph_lowlink(const vector<vector<Int>>& g) {
    Int n = SIZE(g);
    vector<Int> ord(n, -1);
    vector<Int> low(n, -1);

    vector<Int>           articulations;
    vector<pair<Int,Int>> bridges;

    auto dfs = FIX([&g,&ord,&low,&articulations,&bridges](auto&& self, Int v, Int parent, Int k) -> void {
        low[v] = ord[v] = k;

        bool arti = false;
        Int n_child = 0;
        for(Int to : g[v]) {
            // 親または後退辺
            if(ord[to] != -1) {
                if(to != parent)
                    chmin(low[v], ord[to]);
                continue;
            }

            // 子を辿り、low[v] を更新
            ++n_child;
            self(to, v, k+1);
            chmin(low[v], low[to]);

            // 関節点判定(根でない場合)
            if(parent != -1 && low[to] >= ord[v])
                arti = true;

            // 橋判定
            if(low[to] > ord[v])
                bridges.emplace_back(minmax(v,to));
        }
        // 関節点判定(根の場合)
        if(parent == -1 && n_child > 1)
            arti = true;

        if(arti)
            articulations.emplace_back(v);
    });
    dfs(0, -1, 0);

    return make_tuple(articulations, bridges);
}

// 各頂点の (indegree,outdegree) のリストを返す (隣接リスト版)
vector<pair<Int,Int>> graph_degrees_list(const vector<vector<Int>>& g) {
    Int n = SIZE(g);
    vector<pair<Int,Int>> res(n, {0,0});

    REP(from, n) {
        for(Int to : g[from]) {
            ++res[from].second;
            ++res[to].first;
        }
    }

    return res;
}

// 各頂点の (indegree,outdegree) のリストを返す (隣接行列版)
vector<pair<Int,Int>> graph_degrees_matrix(const vector<vector<Int>>& g) {
    Int n = SIZE(g);
    vector<pair<Int,Int>> res(n, {0,0});

    REP(from, n) REP(to, n) {
        Int k = g[from][to];
        res[from].second += k;
        res[to].first    += k;
    }

    return res;
}

// グラフのオイラー路 (隣接リスト版)
//
// g は破壊される
// start: 始点
// digraph: 有向グラフか?
vector<Int> graph_euler_trail_list(vector<vector<Int>>& g, Int start, bool digraph) {
    // スタックオーバーフロー回避のため再帰を使わず自前の stack で処理
    enum Action { CALL, RESUME };

    vector<Int> res;

    stack<tuple<Action,Int>> stk;
    stk.emplace(CALL, start);
    while(!stk.empty()) {
        Action act; Int v; tie(act,v) = POP(stk);
        switch(act) {
        case CALL:
            stk.emplace(RESUME, v);
            while(!g[v].empty()) {
                Int to = g[v].back(); g[v].pop_back();
                if(!digraph)
                    g[to].erase(ALL(find, g[to], v));
                stk.emplace(CALL, to);
            }
            break;
        case RESUME:
            res.emplace_back(v);
            break;
        default: ASSERT(false);
        }
    }

    ALL(reverse, res);

    return res;
}

// 無向グラフのオイラー路 (隣接行列版)
//
// g[v][w]: v,w 間の辺の本数 (破壊される)
// start: 始点
// digraph: 有向グラフか?
vector<Int> graph_euler_trail_matrix(vector<vector<Int>>& g, Int start, bool digraph) {
    // スタックオーバーフロー回避のため再帰を使わず自前の stack で処理
    enum Action { CALL, RESUME };

    Int n = SIZE(g);
    vector<Int> res;

    stack<tuple<Action,Int>> stk;
    stk.emplace(CALL, start);
    while(!stk.empty()) {
        Action act; Int v; tie(act,v) = POP(stk);
        switch(act) {
        case CALL:
            stk.emplace(RESUME, v);
            REP(to, n) {
                if(g[v][to] == 0) continue;
                --g[v][to];
                if(!digraph)
                    --g[to][v];
                stk.emplace(CALL, to);
            }
            break;
        case RESUME:
            res.emplace_back(v);
            break;
        default: ASSERT(false);
        }
    }

    ALL(reverse, res);

    return res;
}

// }}}

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

[[noreturn]] void impossible() {
    PRINTLN("-1");
    EXIT();
}



void solve() {
    Int N = RD();
    Int M = RD();
    auto A = RD_VEC(N);
    auto E = RD_VEC<tuple<Int,Int,Int>>(N-1);

    auto G = wudgraph_list(N, E);

    // 部分木vで(子i個目までで)時間t消費したときの最大スコア
    auto dp = vec_make<Int>(N,M+1, 0);

    auto dfs = FIX([&](auto&& self, Int v, Int p) -> void {
        dp[v][0] = A[v];

        for(const auto& e : G[v]) {
            Int to,cost; tie(to,cost) = e;
            if(to == p) continue;

            self(to, v);
            for(Int t = M; t >= 0; --t) {
                // i個目の子で時間tt消費
                REP(tt, M+1) {
                    Int t_total = t + 2*cost + tt;
                    if(t_total > M) break;
                    chmax(dp[v][t_total], dp[v][t] + dp[to][tt]);
                }
            }
        }
    });
    dfs(0, -1);
    DBG_GRID(dp);

    auto ans = *ALL(max_element, dp[0]);

    PRINTLN(ans);
}

signed main() {
    Int T = 1; //RD();
    LOOP(T) {
        solve();
    }

    EXIT();
}