#line 1 "/Users/nok0/Documents/Programming/nok0/atcoder/math.hpp"



#include <algorithm>
#include <cassert>
#include <tuple>
#include <vector>

#line 1 "/Users/nok0/Documents/Programming/nok0/atcoder/internal_math.hpp"



#include <utility>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
    x %= m;
    if (x < 0) x += m;
    return x;
}

// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
    unsigned int _m;
    unsigned long long im;

    // @param m `1 <= m < 2^31`
    explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}

    // @return m
    unsigned int umod() const { return _m; }

    // @param a `0 <= a < m`
    // @param b `0 <= b < m`
    // @return `a * b % m`
    unsigned int mul(unsigned int a, unsigned int b) const {
        // [1] m = 1
        // a = b = im = 0, so okay

        // [2] m >= 2
        // im = ceil(2^64 / m)
        // -> im * m = 2^64 + r (0 <= r < m)
        // let z = a*b = c*m + d (0 <= c, d < m)
        // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
        // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
        // ((ab * im) >> 64) == c or c + 1
        unsigned long long z = a;
        z *= b;
#ifdef _MSC_VER
        unsigned long long x;
        _umul128(z, im, &x);
#else
        unsigned long long x =
            (unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
        unsigned int v = (unsigned int)(z - x * _m);
        if (_m <= v) v += _m;
        return v;
    }
};

// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
    if (m == 1) return 0;
    unsigned int _m = (unsigned int)(m);
    unsigned long long r = 1;
    unsigned long long y = safe_mod(x, m);
    while (n) {
        if (n & 1) r = (r * y) % _m;
        y = (y * y) % _m;
        n >>= 1;
    }
    return r;
}

// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
constexpr bool is_prime_constexpr(int n) {
    if (n <= 1) return false;
    if (n == 2 || n == 7 || n == 61) return true;
    if (n % 2 == 0) return false;
    long long d = n - 1;
    while (d % 2 == 0) d /= 2;
    constexpr long long bases[3] = {2, 7, 61};
    for (long long a : bases) {
        long long t = d;
        long long y = pow_mod_constexpr(a, t, n);
        while (t != n - 1 && y != 1 && y != n - 1) {
            y = y * y % n;
            t <<= 1;
        }
        if (y != n - 1 && t % 2 == 0) {
            return false;
        }
    }
    return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);

// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
    a = safe_mod(a, b);
    if (a == 0) return {b, 0};

    // Contracts:
    // [1] s - m0 * a = 0 (mod b)
    // [2] t - m1 * a = 0 (mod b)
    // [3] s * |m1| + t * |m0| <= b
    long long s = b, t = a;
    long long m0 = 0, m1 = 1;

    while (t) {
        long long u = s / t;
        s -= t * u;
        m0 -= m1 * u;  // |m1 * u| <= |m1| * s <= b

        // [3]:
        // (s - t * u) * |m1| + t * |m0 - m1 * u|
        // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
        // = s * |m1| + t * |m0| <= b

        auto tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    // by [3]: |m0| <= b/g
    // by g != b: |m0| < b/g
    if (m0 < 0) m0 += b / s;
    return {s, m0};
}

// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
constexpr int primitive_root_constexpr(int m) {
    if (m == 2) return 1;
    if (m == 167772161) return 3;
    if (m == 469762049) return 3;
    if (m == 754974721) return 11;
    if (m == 998244353) return 3;
    int divs[20] = {};
    divs[0] = 2;
    int cnt = 1;
    int x = (m - 1) / 2;
    while (x % 2 == 0) x /= 2;
    for (int i = 3; (long long)(i)*i <= x; i += 2) {
        if (x % i == 0) {
            divs[cnt++] = i;
            while (x % i == 0) {
                x /= i;
            }
        }
    }
    if (x > 1) {
        divs[cnt++] = x;
    }
    for (int g = 2;; g++) {
        bool ok = true;
        for (int i = 0; i < cnt; i++) {
            if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
                ok = false;
                break;
            }
        }
        if (ok) return g;
    }
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

// @param n `n < 2^32`
// @param m `1 <= m < 2^32`
// @return sum_{i=0}^{n-1} floor((ai + b) / m) (mod 2^64)
unsigned long long floor_sum_unsigned(unsigned long long n,
                                      unsigned long long m,
                                      unsigned long long a,
                                      unsigned long long b) {
    unsigned long long ans = 0;
    while (true) {
        if (a >= m) {
            ans += n * (n - 1) / 2 * (a / m);
            a %= m;
        }
        if (b >= m) {
            ans += n * (b / m);
            b %= m;
        }

        unsigned long long y_max = a * n + b;
        if (y_max < m) break;
        // y_max < m * (n + 1)
        // floor(y_max / m) <= n
        n = (unsigned long long)(y_max / m);
        b = (unsigned long long)(y_max % m);
        std::swap(m, a);
    }
    return ans;
}

}  // namespace internal

}  // namespace atcoder


#line 10 "/Users/nok0/Documents/Programming/nok0/atcoder/math.hpp"

namespace atcoder {

long long pow_mod(long long x, long long n, int m) {
    assert(0 <= n && 1 <= m);
    if (m == 1) return 0;
    internal::barrett bt((unsigned int)(m));
    unsigned int r = 1, y = (unsigned int)(internal::safe_mod(x, m));
    while (n) {
        if (n & 1) r = bt.mul(r, y);
        y = bt.mul(y, y);
        n >>= 1;
    }
    return r;
}

long long inv_mod(long long x, long long m) {
    assert(1 <= m);
    auto z = internal::inv_gcd(x, m);
    assert(z.first == 1);
    return z.second;
}

// (rem, mod)
std::pair<long long, long long> crt(const std::vector<long long>& r,
                                    const std::vector<long long>& m) {
    assert(r.size() == m.size());
    int n = int(r.size());
    // Contracts: 0 <= r0 < m0
    long long r0 = 0, m0 = 1;
    for (int i = 0; i < n; i++) {
        assert(1 <= m[i]);
        long long r1 = internal::safe_mod(r[i], m[i]), m1 = m[i];
        if (m0 < m1) {
            std::swap(r0, r1);
            std::swap(m0, m1);
        }
        if (m0 % m1 == 0) {
            if (r0 % m1 != r1) return {0, 0};
            continue;
        }
        // assume: m0 > m1, lcm(m0, m1) >= 2 * max(m0, m1)

        // (r0, m0), (r1, m1) -> (r2, m2 = lcm(m0, m1));
        // r2 % m0 = r0
        // r2 % m1 = r1
        // -> (r0 + x*m0) % m1 = r1
        // -> x*u0*g = r1-r0 (mod u1*g) (u0*g = m0, u1*g = m1)
        // -> x = (r1 - r0) / g * inv(u0) (mod u1)

        // im = inv(u0) (mod u1) (0 <= im < u1)
        long long g, im;
        std::tie(g, im) = internal::inv_gcd(m0, m1);

        long long u1 = (m1 / g);
        // |r1 - r0| < (m0 + m1) <= lcm(m0, m1)
        if ((r1 - r0) % g) return {0, 0};

        // u1 * u1 <= m1 * m1 / g / g <= m0 * m1 / g = lcm(m0, m1)
        long long x = (r1 - r0) / g % u1 * im % u1;

        // |r0| + |m0 * x|
        // < m0 + m0 * (u1 - 1)
        // = m0 + m0 * m1 / g - m0
        // = lcm(m0, m1)
        r0 += x * m0;
        m0 *= u1;  // -> lcm(m0, m1)
        if (r0 < 0) r0 += m0;
    }
    return {r0, m0};
}

long long floor_sum(long long n, long long m, long long a, long long b) {
    assert(0 <= n && n < (1LL << 32));
    assert(1 <= m && m < (1LL << 32));
    unsigned long long ans = 0;
    if (a < 0) {
        unsigned long long a2 = internal::safe_mod(a, m);
        ans -= 1ULL * n * (n - 1) / 2 * ((a2 - a) / m);
        a = a2;
    }
    if (b < 0) {
        unsigned long long b2 = internal::safe_mod(b, m);
        ans -= 1ULL * n * ((b2 - b) / m);
        b = b2;
    }
    return ans + internal::floor_sum_unsigned(n, m, a, b);
}

}  // namespace atcoder


#line 1 "/Users/nok0/Documents/Programming/nok0/cftemp.hpp"
#include <bits/stdc++.h>
using namespace std;

#pragma region Macros
// rep macro
#define foa(v, a) for(auto &v : a)
#define REPname(a, b, c, d, e, ...) e
#define REP(...) REPname(__VA_ARGS__, REP3, REP2, REP1, REP0)(__VA_ARGS__)
#define REP0(x) for(int i = 0; i < (x); ++i)
#define REP1(i, x) for(int i = 0; i < (x); ++i)
#define REP2(i, l, r) for(int i = (l); i < (r); ++i)
#define REP3(i, l, r, c) for(int i = (l); i < (r); i += (c))
#define REPSname(a, b, c, ...) c
#define REPS(...) REPSname(__VA_ARGS__, REPS1, REPS0)(__VA_ARGS__)
#define REPS0(x) for(int i = 1; i <= (x); ++i)
#define REPS1(i, x) for(int i = 1; i <= (x); ++i)
#define RREPname(a, b, c, d, e, ...) e
#define RREP(...) RREPname(__VA_ARGS__, RREP3, RREP2, RREP1, RREP0)(__VA_ARGS__)
#define RREP0(x) for(int i = (x)-1; i >= 0; --i)
#define RREP1(i, x) for(int i = (x)-1; i >= 0; --i)
#define RREP2(i, r, l) for(int i = (r)-1; i >= (l); --i)
#define RREP3(i, r, l, c) for(int i = (r)-1; i >= (l); i -= (c))
#define RREPSname(a, b, c, ...) c
#define RREPS(...) RREPSname(__VA_ARGS__, RREPS1, RREPS0)(__VA_ARGS__)
#define RREPS0(x) for(int i = (x); i >= 1; --i)
#define RREPS1(i, x) for(int i = (x); i >= 1; --i)

// name macro
#define pb push_back
#define eb emplace_back
#define SZ(x) ((int)(x).size())
#define all(x) (x).begin(), (x).end()
#define rall(x) (x).rbegin(), (x).rend()
#define popcnt(x) __builtin_popcountll(x)
template <class T = int>
using V = std::vector<T>;
template <class T = int>
using VV = std::vector<std::vector<T>>;
template <class T>
using pqup = std::priority_queue<T, std::vector<T>, std::greater<T>>;
using ll = long long;
using ld = long double;
using int128 = __int128_t;
using pii = std::pair<int, int>;
using pll = std::pair<long long, long long>;

// input macro
template <class T, class U>
std::istream &operator>>(std::istream &is, std::pair<T, U> &p) {
	is >> p.first >> p.second;
	return is;
}
template <class T>
std::istream &operator>>(std::istream &is, std::vector<T> &v) {
	for(T &i : v) is >> i;
	return is;
}
std::istream &operator>>(std::istream &is, __int128_t &a) {
	std::string s;
	is >> s;
	__int128_t ret = 0;
	for(int i = 0; i < s.length(); i++)
		if('0' <= s[i] and s[i] <= '9')
			ret = 10 * ret + s[i] - '0';
	a = ret * (s[0] == '-' ? -1 : 1);
	return is;
}
namespace scanner {
void scan(int &a) { std::cin >> a; }
void scan(long long &a) { std::cin >> a; }
void scan(std::string &a) { std::cin >> a; }
void scan(char &a) { std::cin >> a; }
void scan(char a[]) { std::scanf("%s", a); }
void scan(double &a) { std::cin >> a; }
void scan(long double &a) { std::cin >> a; }
template <class T, class U>
void scan(std::pair<T, U> &p) { std::cin >> p; }
template <class T>
void scan(std::vector<T> &a) { std::cin >> a; }
void INPUT() {}
template <class Head, class... Tail>
void INPUT(Head &head, Tail &...tail) {
	scan(head);
	INPUT(tail...);
}
}  // namespace scanner
#define VEC(type, name, size)     \
	std::vector<type> name(size); \
	scanner::INPUT(name)
#define VVEC(type, name, h, w)                                    \
	std::vector<std::vector<type>> name(h, std::vector<type>(w)); \
	scanner::INPUT(name)
#define INT(...)     \
	int __VA_ARGS__; \
	scanner::INPUT(__VA_ARGS__)
#define LL(...)            \
	long long __VA_ARGS__; \
	scanner::INPUT(__VA_ARGS__)
#define STR(...)             \
	std::string __VA_ARGS__; \
	scanner::INPUT(__VA_ARGS__)
#define CHAR(...)     \
	char __VA_ARGS__; \
	scanner::INPUT(__VA_ARGS__)
#define DOUBLE(...)     \
	double __VA_ARGS__; \
	scanner::INPUT(__VA_ARGS__)
#define LD(...)              \
	long double __VA_ARGS__; \
	scanner::INPUT(__VA_ARGS__)

// output-macro
template <class T, class U>
std::ostream &operator<<(std::ostream &os, const std::pair<T, U> &p) {
	os << p.first << " " << p.second;
	return os;
}
template <class T>
std::ostream &operator<<(std::ostream &os, const std::vector<T> &a) {
	for(int i = 0; i < int(a.size()); ++i) {
		if(i) os << " ";
		os << a[i];
	}
	return os;
}
std::ostream &operator<<(std::ostream &dest, __int128_t &value) {
	std::ostream::sentry s(dest);
	if(s) {
		__uint128_t tmp = value < 0 ? -value : value;
		char buffer[128];
		char *d = std::end(buffer);
		do {
			--d;
			*d = "0123456789"[tmp % 10];
			tmp /= 10;
		} while(tmp != 0);
		if(value < 0) {
			--d;
			*d = '-';
		}
		int len = std::end(buffer) - d;
		if(dest.rdbuf()->sputn(d, len) != len) {
			dest.setstate(std::ios_base::badbit);
		}
	}
	return dest;
}
template <class T>
void print(const T a) { std::cout << a << '\n'; }
template <class Head, class... Tail>
void print(Head H, Tail... T) {
	std::cout << H << ' ';
	print(T...);
}
template <class T>
void printel(const T a) { std::cout << a << '\n'; }
template <class T>
void printel(const std::vector<T> &a) {
	for(const auto &v : a)
		std::cout << v << '\n';
}
template <class Head, class... Tail>
void printel(Head H, Tail... T) {
	std::cout << H << '\n';
	printel(T...);
}
void Yes(const bool b = true) { std::cout << (b ? "Yes\n" : "No\n"); }
void No() { std::cout << "No\n"; }
void YES(const bool b = true) { std::cout << (b ? "YES\n" : "NO\n"); }
void NO() { std::cout << "NO\n"; }
void err(const bool b = true) {
	if(b) {
		std::cout << "-1\n", exit(0);
	}
}

//debug macro
namespace debugger {
template <class T>
void view(const std::vector<T> &a) {
	std::cerr << "{ ";
	for(const auto &v : a) {
		std::cerr << v << ", ";
	}
	std::cerr << "\b\b }";
}
template <class T>
void view(const std::vector<std::vector<T>> &a) {
	std::cerr << "{\n";
	for(const auto &v : a) {
		std::cerr << "\t";
		view(v);
		std::cerr << "\n";
	}
	std::cerr << "}";
}
template <class T, class U>
void view(const std::vector<std::pair<T, U>> &a) {
	std::cerr << "{\n";
	for(const auto &p : a) std::cerr << "\t(" << p.first << ", " << p.second << ")\n";
	std::cerr << "}";
}
template <class T, class U>
void view(const std::map<T, U> &m) {
	std::cerr << "{\n";
	for(const auto &p : m) std::cerr << "\t[" << p.first << "] : " << p.second << "\n";
	std::cerr << "}";
}
template <class T, class U>
void view(const std::pair<T, U> &p) { std::cerr << "(" << p.first << ", " << p.second << ")"; }
template <class T>
void view(const std::set<T> &s) {
	std::cerr << "{ ";
	for(auto &v : s) {
		view(v);
		std::cerr << ", ";
	}
	std::cerr << "\b\b }";
}

template <class T>
void view(const T &e) { std::cerr << e; }
}  // namespace debugger
#ifdef LOCAL
void debug_out() {}
template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
	debugger::view(H);
	std::cerr << ", ";
	debug_out(T...);
}
#define debug(...)                                                \
	do {                                                          \
		std::cerr << __LINE__ << " [" << #__VA_ARGS__ << "] : ["; \
		debug_out(__VA_ARGS__);                                   \
		std::cerr << "\b\b]\n";                                   \
	} while(false)
#else
#define debug(...) (void(0))
#endif

// vector macro
template <class T>
int lb(const std::vector<T> &a, const T x) { return std::distance((a).begin(), std::lower_bound((a).begin(), (a).end(), (x))); }
template <class T>
int ub(const std::vector<T> &a, const T x) { return std::distance((a).begin(), std::upper_bound((a).begin(), (a).end(), (x))); }
template <class T>
void UNIQUE(std::vector<T> &a) {
	std::sort(a.begin(), a.end());
	a.erase(std::unique(a.begin(), a.end()), a.end());
}
template <class T>
std::vector<T> press(std::vector<T> &a) {
	auto res = a;
	UNIQUE(res);
	for(auto &v : a)
		v = lb(res, v);
	return res;
}
#define SORTname(a, b, c, ...) c
#define SORT(...) SORTname(__VA_ARGS__, SORT1, SORT0, ...)(__VA_ARGS__)
#define SORT0(a) std::sort((a).begin(), (a).end())
#define SORT1(a, c) std::sort((a).begin(), (a).end(), [](const auto x, const auto y) { return x c y; })
template <class T>
void ADD(std::vector<T> &a, const T x = 1) {
	for(auto &v : a) v += x;
}
template <class T>
void SUB(std::vector<T> &a, const T x = 1) {
	for(auto &v : a) v -= x;
}
std::vector<std::pair<char, int>> rle(const string &s) {
	int n = s.size();
	std::vector<std::pair<char, int>> ret;
	for(int l = 0; l < n;) {
		int r = l + 1;
		for(; r < n and s[l] == s[r]; r++) {}
		ret.emplace_back(s[l], r - l);
		l = r;
	}
	return ret;
}
template <class T>
std::vector<std::pair<T, int>> rle(const std::vector<T> &v) {
	int n = v.size();
	std::vector<std::pair<T, int>> ret;
	for(int l = 0; l < n;) {
		int r = l + 1;
		for(; r < n and v[l] == v[r]; r++) {}
		ret.emplace_back(v[l], r - l);
		l = r;
	}
	return ret;
}
std::vector<int> iota(int n) {
	std::vector<int> p(n);
	std::iota(p.begin(), p.end(), 0);
	return p;
}
template <class T>
struct cum_vector {
public:
	cum_vector() = default;
	template <class U>
	cum_vector(const std::vector<U> &vec) : cum((int)vec.size() + 1) {
		for(int i = 0; i < (int)vec.size(); i++)
			cum[i + 1] = cum[i] + vec[i];
	}
	T prod(int l, int r) {
		return cum[r] - cum[l];
	}

private:
	std::vector<T> cum;
};

// math macro
template <class T, class U>
inline bool chmin(T &a, const U &b) { return a > b ? a = b, true : false; }
template <class T, class U>
inline bool chmax(T &a, const U &b) { return a < b ? a = b, true : false; }
template <class T>
T divup(T x, T y) { return (x + y - 1) / y; }
template <class T>
T POW(T a, long long n) {
	T ret = 1;
	while(n) {
		if(n & 1) ret *= a;
		a *= a;
		n >>= 1;
	}
	return ret;
}
// modpow
long long POW(long long a, long long n, const int mod) {
	long long ret = 1;
	a = (a % mod + mod) % mod;
	while(n) {
		if(n & 1) (ret *= a) %= mod;
		(a *= a) %= mod;
		n >>= 1;
	}
	return ret;
}
template <class T, class F>
T bin_search(T ok, T ng, const F &f) {
	while(abs(ok - ng) > 1) {
		T mid = (ok + ng) >> 1;
		(f(mid) ? ok : ng) = mid;
	}
	return ok;
}
template <class T, class F>
T bin_search(T ok, T ng, const F &f, int loop) {
	for(int i = 0; i < loop; i++) {
		T mid = (ok + ng) / 2;
		(f(mid) ? ok : ng) = mid;
	}
	return ok;
}

// others
struct fast_io {
	fast_io() {
		ios::sync_with_stdio(false);
		cin.tie(nullptr);
		cout << fixed << setprecision(15);
	}
} fast_io_;
const int inf = 1e9;
const ll INF = 1e18;
#pragma endregion

void main_();

int main() {
	main_();
	return 0;
}
#line 7 "/Users/nok0/Documents/Programming/nok0/math/formal_power_series.hpp"

#line 1 "/Users/nok0/Documents/Programming/nok0/atcoder/convolution.hpp"



#line 7 "/Users/nok0/Documents/Programming/nok0/atcoder/convolution.hpp"
#include <type_traits>
#line 9 "/Users/nok0/Documents/Programming/nok0/atcoder/convolution.hpp"

#line 1 "/Users/nok0/Documents/Programming/nok0/atcoder/internal_bit.hpp"



#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

// @param n `0 <= n`
// @return minimum non-negative `x` s.t. `n <= 2**x`
int ceil_pow2(int n) {
    int x = 0;
    while ((1U << x) < (unsigned int)(n)) x++;
    return x;
}

// @param n `1 <= n`
// @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`
constexpr int bsf_constexpr(unsigned int n) {
    int x = 0;
    while (!(n & (1 << x))) x++;
    return x;
}

// @param n `1 <= n`
// @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`
int bsf(unsigned int n) {
#ifdef _MSC_VER
    unsigned long index;
    _BitScanForward(&index, n);
    return index;
#else
    return __builtin_ctz(n);
#endif
}

}  // namespace internal

}  // namespace atcoder


#line 1 "/Users/nok0/Documents/Programming/nok0/atcoder/modint.hpp"



#line 7 "/Users/nok0/Documents/Programming/nok0/atcoder/modint.hpp"

#ifdef _MSC_VER
#include <intrin.h>
#endif

#line 1 "/Users/nok0/Documents/Programming/nok0/atcoder/internal_type_traits.hpp"



#line 7 "/Users/nok0/Documents/Programming/nok0/atcoder/internal_type_traits.hpp"

namespace atcoder {

namespace internal {

#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value ||
                                  std::is_same<T, __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using is_unsigned_int128 =
    typename std::conditional<std::is_same<T, __uint128_t>::value ||
                                  std::is_same<T, unsigned __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using make_unsigned_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value,
                              __uint128_t,
                              unsigned __int128>;

template <class T>
using is_integral = typename std::conditional<std::is_integral<T>::value ||
                                                  is_signed_int128<T>::value ||
                                                  is_unsigned_int128<T>::value,
                                              std::true_type,
                                              std::false_type>::type;

template <class T>
using is_signed_int = typename std::conditional<(is_integral<T>::value &&
                                                 std::is_signed<T>::value) ||
                                                    is_signed_int128<T>::value,
                                                std::true_type,
                                                std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<(is_integral<T>::value &&
                               std::is_unsigned<T>::value) ||
                                  is_unsigned_int128<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<
    is_signed_int128<T>::value,
    make_unsigned_int128<T>,
    typename std::conditional<std::is_signed<T>::value,
                              std::make_unsigned<T>,
                              std::common_type<T>>::type>::type;

#else

template <class T> using is_integral = typename std::is_integral<T>;

template <class T>
using is_signed_int =
    typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<is_integral<T>::value &&
                                  std::is_unsigned<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<is_signed_int<T>::value,
                                              std::make_unsigned<T>,
                                              std::common_type<T>>::type;

#endif

template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;

template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;

template <class T> using to_unsigned_t = typename to_unsigned<T>::type;

}  // namespace internal

}  // namespace atcoder


#line 14 "/Users/nok0/Documents/Programming/nok0/atcoder/modint.hpp"

namespace atcoder {

namespace internal {

struct modint_base {};
struct static_modint_base : modint_base {};

template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;

}  // namespace internal

template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
struct static_modint : internal::static_modint_base {
    using mint = static_modint;

  public:
    static constexpr int mod() { return m; }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    static_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    static_modint(T v) {
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0) x += umod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    static_modint(T v) {
        _v = (unsigned int)(v % umod());
    }

    unsigned int val() const { return _v; }

    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v -= rhs._v;
        if (_v >= umod()) _v += umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        unsigned long long z = _v;
        z *= rhs._v;
        _v = (unsigned int)(z % umod());
        return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        if (prime) {
            assert(_v);
            return pow(umod() - 2);
        } else {
            auto eg = internal::inv_gcd(_v, m);
            assert(eg.first == 1);
            return eg.second;
        }
    }

    friend mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static constexpr unsigned int umod() { return m; }
    static constexpr bool prime = internal::is_prime<m>;
};

template <int id> struct dynamic_modint : internal::modint_base {
    using mint = dynamic_modint;

  public:
    static int mod() { return (int)(bt.umod()); }
    static void set_mod(int m) {
        assert(1 <= m);
        bt = internal::barrett(m);
    }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    dynamic_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    dynamic_modint(T v) {
        long long x = (long long)(v % (long long)(mod()));
        if (x < 0) x += mod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    dynamic_modint(T v) {
        _v = (unsigned int)(v % mod());
    }

    unsigned int val() const { return _v; }

    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v += mod() - rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        _v = bt.mul(_v, rhs._v);
        return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        auto eg = internal::inv_gcd(_v, mod());
        assert(eg.first == 1);
        return eg.second;
    }

    friend mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static internal::barrett bt;
    static unsigned int umod() { return bt.umod(); }
};
template <int id> internal::barrett dynamic_modint<id>::bt(998244353);

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;

namespace internal {

template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;

template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;

template <class> struct is_dynamic_modint : public std::false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};

template <class T>
using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;

}  // namespace internal

}  // namespace atcoder


#line 12 "/Users/nok0/Documents/Programming/nok0/atcoder/convolution.hpp"

namespace atcoder {

namespace internal {

template <class mint,
          int g = internal::primitive_root<mint::mod()>,
          internal::is_static_modint_t<mint>* = nullptr>
struct fft_info {
    static constexpr int rank2 = bsf_constexpr(mint::mod() - 1);
    std::array<mint, rank2 + 1> root;   // root[i]^(2^i) == 1
    std::array<mint, rank2 + 1> iroot;  // root[i] * iroot[i] == 1

    std::array<mint, std::max(0, rank2 - 2 + 1)> rate2;
    std::array<mint, std::max(0, rank2 - 2 + 1)> irate2;

    std::array<mint, std::max(0, rank2 - 3 + 1)> rate3;
    std::array<mint, std::max(0, rank2 - 3 + 1)> irate3;

    fft_info() {
        root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2);
        iroot[rank2] = root[rank2].inv();
        for (int i = rank2 - 1; i >= 0; i--) {
            root[i] = root[i + 1] * root[i + 1];
            iroot[i] = iroot[i + 1] * iroot[i + 1];
        }

        {
            mint prod = 1, iprod = 1;
            for (int i = 0; i <= rank2 - 2; i++) {
                rate2[i] = root[i + 2] * prod;
                irate2[i] = iroot[i + 2] * iprod;
                prod *= iroot[i + 2];
                iprod *= root[i + 2];
            }
        }
        {
            mint prod = 1, iprod = 1;
            for (int i = 0; i <= rank2 - 3; i++) {
                rate3[i] = root[i + 3] * prod;
                irate3[i] = iroot[i + 3] * iprod;
                prod *= iroot[i + 3];
                iprod *= root[i + 3];
            }
        }
    }
};

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly(std::vector<mint>& a) {
    int n = int(a.size());
    int h = internal::ceil_pow2(n);

    static const fft_info<mint> info;

    int len = 0;  // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
    while (len < h) {
        if (h - len == 1) {
            int p = 1 << (h - len - 1);
            mint rot = 1;
            for (int s = 0; s < (1 << len); s++) {
                int offset = s << (h - len);
                for (int i = 0; i < p; i++) {
                    auto l = a[i + offset];
                    auto r = a[i + offset + p] * rot;
                    a[i + offset] = l + r;
                    a[i + offset + p] = l - r;
                }
                if (s + 1 != (1 << len))
                    rot *= info.rate2[bsf(~(unsigned int)(s))];
            }
            len++;
        } else {
            // 4-base
            int p = 1 << (h - len - 2);
            mint rot = 1, imag = info.root[2];
            for (int s = 0; s < (1 << len); s++) {
                mint rot2 = rot * rot;
                mint rot3 = rot2 * rot;
                int offset = s << (h - len);
                for (int i = 0; i < p; i++) {
                    auto mod2 = 1ULL * mint::mod() * mint::mod();
                    auto a0 = 1ULL * a[i + offset].val();
                    auto a1 = 1ULL * a[i + offset + p].val() * rot.val();
                    auto a2 = 1ULL * a[i + offset + 2 * p].val() * rot2.val();
                    auto a3 = 1ULL * a[i + offset + 3 * p].val() * rot3.val();
                    auto a1na3imag =
                        1ULL * mint(a1 + mod2 - a3).val() * imag.val();
                    auto na2 = mod2 - a2;
                    a[i + offset] = a0 + a2 + a1 + a3;
                    a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
                    a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
                    a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
                }
                if (s + 1 != (1 << len))
                    rot *= info.rate3[bsf(~(unsigned int)(s))];
            }
            len += 2;
        }
    }
}

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly_inv(std::vector<mint>& a) {
    int n = int(a.size());
    int h = internal::ceil_pow2(n);

    static const fft_info<mint> info;

    int len = h;  // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
    while (len) {
        if (len == 1) {
            int p = 1 << (h - len);
            mint irot = 1;
            for (int s = 0; s < (1 << (len - 1)); s++) {
                int offset = s << (h - len + 1);
                for (int i = 0; i < p; i++) {
                    auto l = a[i + offset];
                    auto r = a[i + offset + p];
                    a[i + offset] = l + r;
                    a[i + offset + p] =
                        (unsigned long long)(mint::mod() + l.val() - r.val()) *
                        irot.val();
                    ;
                }
                if (s + 1 != (1 << (len - 1)))
                    irot *= info.irate2[bsf(~(unsigned int)(s))];
            }
            len--;
        } else {
            // 4-base
            int p = 1 << (h - len);
            mint irot = 1, iimag = info.iroot[2];
            for (int s = 0; s < (1 << (len - 2)); s++) {
                mint irot2 = irot * irot;
                mint irot3 = irot2 * irot;
                int offset = s << (h - len + 2);
                for (int i = 0; i < p; i++) {
                    auto a0 = 1ULL * a[i + offset + 0 * p].val();
                    auto a1 = 1ULL * a[i + offset + 1 * p].val();
                    auto a2 = 1ULL * a[i + offset + 2 * p].val();
                    auto a3 = 1ULL * a[i + offset + 3 * p].val();

                    auto a2na3iimag =
                        1ULL *
                        mint((mint::mod() + a2 - a3) * iimag.val()).val();

                    a[i + offset] = a0 + a1 + a2 + a3;
                    a[i + offset + 1 * p] =
                        (a0 + (mint::mod() - a1) + a2na3iimag) * irot.val();
                    a[i + offset + 2 * p] =
                        (a0 + a1 + (mint::mod() - a2) + (mint::mod() - a3)) *
                        irot2.val();
                    a[i + offset + 3 * p] =
                        (a0 + (mint::mod() - a1) + (mint::mod() - a2na3iimag)) *
                        irot3.val();
                }
                if (s + 1 != (1 << (len - 2)))
                    irot *= info.irate3[bsf(~(unsigned int)(s))];
            }
            len -= 2;
        }
    }
}

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution_naive(const std::vector<mint>& a,
                                    const std::vector<mint>& b) {
    int n = int(a.size()), m = int(b.size());
    std::vector<mint> ans(n + m - 1);
    if (n < m) {
        for (int j = 0; j < m; j++) {
            for (int i = 0; i < n; i++) {
                ans[i + j] += a[i] * b[j];
            }
        }
    } else {
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                ans[i + j] += a[i] * b[j];
            }
        }
    }
    return ans;
}

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution_fft(std::vector<mint> a, std::vector<mint> b) {
    int n = int(a.size()), m = int(b.size());
    int z = 1 << internal::ceil_pow2(n + m - 1);
    a.resize(z);
    internal::butterfly(a);
    b.resize(z);
    internal::butterfly(b);
    for (int i = 0; i < z; i++) {
        a[i] *= b[i];
    }
    internal::butterfly_inv(a);
    a.resize(n + m - 1);
    mint iz = mint(z).inv();
    for (int i = 0; i < n + m - 1; i++) a[i] *= iz;
    return a;
}

}  // namespace internal

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution(std::vector<mint>&& a, std::vector<mint>&& b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};
    if (std::min(n, m) <= 60) return convolution_naive(a, b);
    return internal::convolution_fft(a, b);
}

template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution(const std::vector<mint>& a,
                              const std::vector<mint>& b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};
    if (std::min(n, m) <= 60) return convolution_naive(a, b);
    return internal::convolution_fft(a, b);
}

template <unsigned int mod = 998244353,
          class T,
          std::enable_if_t<internal::is_integral<T>::value>* = nullptr>
std::vector<T> convolution(const std::vector<T>& a, const std::vector<T>& b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};

    using mint = static_modint<mod>;
    std::vector<mint> a2(n), b2(m);
    for (int i = 0; i < n; i++) {
        a2[i] = mint(a[i]);
    }
    for (int i = 0; i < m; i++) {
        b2[i] = mint(b[i]);
    }
    auto c2 = convolution(move(a2), move(b2));
    std::vector<T> c(n + m - 1);
    for (int i = 0; i < n + m - 1; i++) {
        c[i] = c2[i].val();
    }
    return c;
}

std::vector<long long> convolution_ll(const std::vector<long long>& a,
                                      const std::vector<long long>& b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m) return {};

    static constexpr unsigned long long MOD1 = 754974721;  // 2^24
    static constexpr unsigned long long MOD2 = 167772161;  // 2^25
    static constexpr unsigned long long MOD3 = 469762049;  // 2^26
    static constexpr unsigned long long M2M3 = MOD2 * MOD3;
    static constexpr unsigned long long M1M3 = MOD1 * MOD3;
    static constexpr unsigned long long M1M2 = MOD1 * MOD2;
    static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;

    static constexpr unsigned long long i1 =
        internal::inv_gcd(MOD2 * MOD3, MOD1).second;
    static constexpr unsigned long long i2 =
        internal::inv_gcd(MOD1 * MOD3, MOD2).second;
    static constexpr unsigned long long i3 =
        internal::inv_gcd(MOD1 * MOD2, MOD3).second;

    auto c1 = convolution<MOD1>(a, b);
    auto c2 = convolution<MOD2>(a, b);
    auto c3 = convolution<MOD3>(a, b);

    std::vector<long long> c(n + m - 1);
    for (int i = 0; i < n + m - 1; i++) {
        unsigned long long x = 0;
        x += (c1[i] * i1) % MOD1 * M2M3;
        x += (c2[i] * i2) % MOD2 * M1M3;
        x += (c3[i] * i3) % MOD3 * M1M2;
        // B = 2^63, -B <= x, r(real value) < B
        // (x, x - M, x - 2M, or x - 3M) = r (mod 2B)
        // r = c1[i] (mod MOD1)
        // focus on MOD1
        // r = x, x - M', x - 2M', x - 3M' (M' = M % 2^64) (mod 2B)
        // r = x,
        //     x - M' + (0 or 2B),
        //     x - 2M' + (0, 2B or 4B),
        //     x - 3M' + (0, 2B, 4B or 6B) (without mod!)
        // (r - x) = 0, (0)
        //           - M' + (0 or 2B), (1)
        //           -2M' + (0 or 2B or 4B), (2)
        //           -3M' + (0 or 2B or 4B or 6B) (3) (mod MOD1)
        // we checked that
        //   ((1) mod MOD1) mod 5 = 2
        //   ((2) mod MOD1) mod 5 = 3
        //   ((3) mod MOD1) mod 5 = 4
        long long diff =
            c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));
        if (diff < 0) diff += MOD1;
        static constexpr unsigned long long offset[5] = {
            0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3};
        x -= offset[diff % 5];
        c[i] = x;
    }

    return c;
}

}  // namespace atcoder


#line 4 "/Users/nok0/Documents/Programming/nok0/math/factorial.hpp"

#line 6 "/Users/nok0/Documents/Programming/nok0/math/factorial.hpp"

template <class T>
struct factorial {
public:
	static int MAX;
	static std::vector<T> fac, finv, inv;

	factorial() {}

	T binom(int n, int r) {
		if(n < r or n < 0 or r < 0) return T(0);
		assert(n < MAX);
		return fac[n] * finv[r] * finv[n - r];
	}

	T large_binom(int n, int r) {
		if(n < r or n < 0 or r < 0) return T(0);
		assert(r < MAX);
		T ret = finv[r];
		for(int i = 1; i <= r; ++i)
			ret *= (n + 1 - i);
		return ret;
	}

	static void set_size(int n = 3000000) {
		MAX = (n > 1 ? n : 1) + 1;
		if((int)fac.size() >= MAX) return;
		fac.resize(MAX);
		finv.resize(MAX);
		inv.resize(MAX);
		const int MOD = T::mod();
		fac[0] = fac[1] = 1;
		finv[0] = finv[1] = 1;
		inv[1] = 1;
		for(int i = 2; i < MAX; i++) {
			fac[i] = fac[i - 1] * i;
			inv[i] = (T)MOD - inv[MOD % i] * (MOD / i);
			finv[i] = finv[i - 1] * inv[i];
		}
	}
};
template <class T>
int factorial<T>::MAX = 0;
template <class T>
std::vector<T> factorial<T>::fac;
template <class T>
std::vector<T> factorial<T>::finv;
template <class T>
std::vector<T> factorial<T>::inv;
#line 10 "/Users/nok0/Documents/Programming/nok0/math/formal_power_series.hpp"

enum Mode {
	FAST = 1,
	NAIVE = -1,
};
template <class T, Mode mode = FAST>
struct formal_power_series : std::vector<T> {
	factorial<T> fact;
	using std::vector<T>::vector;
	using std::vector<T>::size;
	using std::vector<T>::resize;
	using std::vector<T>::begin;
	using std::vector<T>::insert;
	using std::vector<T>::erase;
	using F = formal_power_series;
	using S = std::vector<std::pair<int, T>>;

	F &operator+=(const F &g) {
		for(int i = 0; i < int(std::min((*this).size(), g.size())); i++) (*this)[i] += g[i];
		return *this;
	}

	F &operator+=(const T &t) {
		assert(int((*this).size()));
		(*this)[0] += t;
		return *this;
	}

	F &operator-=(const F &g) {
		for(int i = 0; i < int(std::min((*this).size(), g.size())); i++) (*this)[i] -= g[i];
		return *this;
	}

	F &operator-=(const T &t) {
		assert(int((*this).size()));
		(*this)[0] -= t;
		return *this;
	}

	F &operator*=(const T &t) {
		for(int i = 0; i < int((*this).size()); ++i) (*this)[i] *= t;
		return *this;
	}

	F &operator/=(const T &t) {
		T div = t.inv();
		for(int i = 0; i < int((*this).size()); ++i) (*this)[i] *= div;
		return *this;
	}

	F &operator>>=(const int sz) {
		assert(sz >= 0);
		int n = (*this).size();
		(*this).erase((*this).begin(), (*this).begin() + std::min(sz, n));
		(*this).resize(n);
		return *this;
	}

	F &operator<<=(const int sz) {
		assert(sz >= 0);
		int n = (*this).size();
		(*this).insert((*this).begin(), sz, T(0));
		(*this).resize(n);
		return *this;
	}

	F poly_div(const F &g) {
		if(this->size() < g.size()) {
			F ret(this->size());
			return ret;
		}
		if(mode == FAST) {
			auto ret = *this;
			int old = this->size();
			int n = old - g.size() + 1;
			ret = ((*this).rev().pre(n) * g.rev().inv(n));
			ret.rev_inplace();
			ret.resize(old);
			return ret;
		} else {
			assert(g.back() != T(0));
			T igb = g.back().inv();
			int n = (*this).size(), m = g.size();
			F this_copy(*this);
			F ret(n);
			for(int i = n - 1; i >= m - 1; --i) {
				T mul = this_copy[i] * igb;
				ret[i - m + 1] = mul;
				for(int j = i; j > i - m; j--)
					this_copy[j] -= g[j - i + m - 1] * mul;
			}
			return ret;
		}
	}

	//これのみ多項式の除算として扱う
	F &operator%=(const F &g) {
		return *this -= this->poly_div(g) * g;
	}

	F &operator=(const std::vector<T> &v) {
		int n = (*this).size();
		for(int i = 0; i < n; ++i) (*this)[i] = v[i];
		return *this;
	}

	F operator-() const {
		F ret = *this;
		return ret * -1;
	}

	F &operator*=(const F &g) {
		if(mode == FAST) {
			int n = (*this).size();
			auto tmp = atcoder::convolution(*this, g);
			for(int i = 0; i < n; ++i) (*this)[i] = tmp[i];
			return *this;
		} else {
			int n = (*this).size(), m = g.size();
			for(int i = n - 1; i >= 0; --i) {
				(*this)[i] *= g[0];
				for(int j = 1; j < std::min(i + 1, m); j++)
					(*this)[i] += (*this)[i - j] * g[j];
			}
			return *this;
		}
	}

	F &operator/=(const F &g) {
		if((*this).size() < g.size()) {
			(*this).assign((*this).size(), T(0));
			return *this;
		}
		if(mode == FAST) {
			*this *= g.inv();
			return *this;
		} else {
			assert(g[0] != T(0));
			T ig0 = g[0].inv();
			int n = (*this).size(), m = g.size();
			for(int i = 0; i < n; ++i) {
				for(int j = 1; j < std::min(i + 1, m); ++j)
					(*this)[i] -= (*this)[i - j] * g[j];
				(*this)[i] *= ig0;
			}
			return *this;
		}
	}

	F &operator*=(S g) {
		int n = (*this).size();
		auto [d, c] = g.front();
		if(!d)
			g.erase(g.begin());
		else
			c = 0;
		for(int i = n - 1; i >= 0; --i) {
			(*this)[i] *= c;
			for(auto &[j, b] : g) {
				if(j > i) break;
				(*this)[i] += (*this)[i - j] * b;
			}
		}
		return *this;
	}

	F &operator/=(S g) {
		int n = (*this).size();
		auto [d, c] = g.front();
		assert(!d and c != 0);
		T ic = c.inv();
		g.erase(g.begin());
		for(int i = 0; i < n; ++i) {
			for(auto &[j, b] : g) {
				if(j > i) break;
				(*this)[i] -= (*this)[i - j] * b;
			}
			(*this)[i] *= ic;
		}
		return *this;
	}

	F operator+(const F &g) const { return F(*this) += g; }

	F operator+(const T &t) const { return F(*this) += t; }

	F operator-(const F &g) const { return F(*this) -= g; }

	F operator-(const T &t) const { return F(*this) -= t; }

	F operator*(const F &g) const { return F(*this) *= g; }

	F operator*(const T &t) const { return F(*this) *= t; }

	F operator/(const F &g) const { return F(*this) /= g; }

	F operator/(const T &t) const { return F(*this) /= t; }

	F operator%(const F &g) const { return F(*this) %= g; }

	F operator*=(const S &g) const { return F(*this) *= g; }

	F operator/=(const S &g) const { return F(*this) /= g; }

	F operator<<(const int sz) const { return F(*this) <<= sz; }

	F operator>>(const int sz) const { return F(*this) >>= sz; }

	F pre(int d) const { return F((*this).begin(), (*this).begin() + std::min((int)(*this).size(), d)); }

	F &shrink() {
		while(!(*this).empty() and (*this).back() == T(0)) (*this).pop_back();
		return *this;
	}

	F &rev_inplace() {
		reverse((*this).begin(), (*this).end());
		return *this;
	}
	F rev() const { return F(*this).rev_inplace(); }

	// *=(1 + cz^d)
	F &multiply(const int d, const T c) {
		int n = (*this).size();
		if(c == T(1))
			for(int i = n - d - 1; i >= 0; --i)
				(*this)[i + d] += (*this)[i];
		else if(c == T(-1))
			for(int i = n - d - 1; i >= 0; --i)
				(*this)[i + d] -= (*this)[i];
		else
			for(int i = n - d - 1; i >= 0; --i)
				(*this)[i + d] += (*this)[i] * c;
		return *this;
	}
	// /=(1 + cz^d)
	F &divide(const int d, const T c) {
		int n = (*this).size();
		if(c == T(1))
			for(int i = 0; i < n - d; ++i) (*this)[i + d] -= (*this)[i];
		else if(c == T(-1))
			for(int i = 0; i < n - d; ++i) (*this)[i + d] += (*this)[i];
		else
			for(int i = 0; i < n - d; ++i) (*this)[i + d] -= (*this)[i] * c;
		return *this;
	}

	//Ο(N)
	T eval(const T &t) const {
		int n = (*this).size();
		T res = 0, tmp = 1;
		for(int i = 0; i < n; ++i) res += (*this)[i] * tmp, tmp *= t;
		return res;
	}

	F inv(int deg = -1) const {
		int n = (*this).size();
		assert(mode == FAST and n and (*this)[0] != 0);
		if(deg == -1) deg = n;
		assert(deg > 0);
		F res{(*this)[0].inv()};
		while(int(res.size()) < deg) {
			int m = res.size();
			F f((*this).begin(), (*this).begin() + std::min(n, m * 2)), r(res);
			f.resize(m * 2), atcoder::internal::butterfly(f);
			r.resize(m * 2), atcoder::internal::butterfly(r);
			for(int i = 0; i < m * 2; ++i) f[i] *= r[i];
			atcoder::internal::butterfly_inv(f);
			f.erase(f.begin(), f.begin() + m);
			f.resize(m * 2), atcoder::internal::butterfly(f);
			for(int i = 0; i < m * 2; ++i) f[i] *= r[i];
			atcoder::internal::butterfly_inv(f);
			T iz = T(m * 2).inv();
			iz *= -iz;
			for(int i = 0; i < m; ++i) f[i] *= iz;
			res.insert(res.end(), f.begin(), f.begin() + m);
		}
		res.resize(deg);
		return res;
	}

	//Ο(N)
	F &diff_inplace() {
		int n = (*this).size();
		for(int i = 1; i < n; ++i) (*this)[i - 1] = (*this)[i] * i;
		(*this)[n - 1] = 0;
		return *this;
	}
	F diff() const { return F(*this).diff_inplace(); }

	//Ο(N)
	F &integral_inplace() {
		int n = (*this).size(), mod = T::mod();
		std::vector<T> inv(n);
		{
			inv[1] = 1;
			for(int i = 2; i < n; ++i)
				inv[i] = T(mod) - inv[mod % i] * (mod / i);
		}
		for(int i = n - 2; i >= 0; --i) (*this)[i + 1] = (*this)[i] * inv[i + 1];
		(*this)[0] = 0;
		return *this;
	}
	F integral() const { return F(*this).integral_inplace(); }

	//Ο(NlogN)
	F &log_inplace() {
		int n = (*this).size();
		assert(n and (*this)[0] == 1);
		F f_inv = (*this).inv();
		(*this).diff_inplace();
		(*this) *= f_inv;
		(*this).integral_inplace();
		return *this;
	}
	F log() const { return F(*this).log_inplace(); }

	//Ο(NlogN)
	F &deriv_inplace() {
		int n = (*this).size();
		assert(n);
		for(int i = 2; i < n; ++i) (*this)[i] *= i;
		(*this).erase((*this).begin());
		(*this).push_back(0);
		return *this;
	}
	F deriv() const { return F(*this).deriv_inplace(); }

	//Ο(NlogN)
	F &exp_inplace() {
		int n = (*this).size();
		assert(n and (*this)[0] == 0);
		F g{1};
		(*this)[0] = 1;
		F h_drv((*this).deriv());
		for(int m = 1; m < n; m *= 2) {
			F f((*this).begin(), (*this).begin() + m);
			f.resize(2 * m), atcoder::internal::butterfly(f);
			auto mult_f = [&](F &p) {
				p.resize(2 * m);
				atcoder::internal::butterfly(p);
				for(int i = 0; i < 2 * m; ++i) p[i] *= f[i];
				atcoder::internal::butterfly_inv(p);
				p /= 2 * m;
			};
			if(m > 1) {
				F g_(g);
				g_.resize(2 * m), atcoder::internal::butterfly(g_);
				for(int i = 0; i < 2 * m; ++i) g_[i] *= g_[i] * f[i];
				atcoder::internal::butterfly_inv(g_);
				T iz = T(-2 * m).inv();
				g_ *= iz;
				g.insert(g.end(), g_.begin() + m / 2, g_.begin() + m);
			}
			F t((*this).begin(), (*this).begin() + m);
			t.deriv_inplace();
			{
				F r{h_drv.begin(), h_drv.begin() + m - 1};
				mult_f(r);
				for(int i = 0; i < m; ++i) t[i] -= r[i] + r[m + i];
			}
			t.insert(t.begin(), t.back());
			t.pop_back();
			t *= g;
			F v((*this).begin() + m, (*this).begin() + std::min(n, 2 * m));
			v.resize(m);
			t.insert(t.begin(), m - 1, 0);
			t.push_back(0);
			t.integral_inplace();
			for(int i = 0; i < m; ++i) v[i] -= t[m + i];
			mult_f(v);
			for(int i = 0; i < std::min(n - m, m); ++i)
				(*this)[m + i] = v[i];
		}
		return *this;
	}
	F exp() const { return F(*this).exp_inplace(); }

	//Ο(NlogN)
	F &pow_inplace(long long k) {
		int n = (*this).size(), l = 0;
		assert(k >= 0);
		if(!k) {
			for(int i = 0; i < n; ++i) (*this)[i] = !i;
			return *this;
		}
		while(l < n and (*this)[l] == 0) ++l;
		if(l > (n - 1) / k or l == n) return *this = F(n);
		T c = (*this)[l];
		(*this).erase((*this).begin(), (*this).begin() + l);
		(*this) /= c;
		(*this).log_inplace();
		(*this).resize(n - l * k);
		(*this) *= k;
		(*this).exp_inplace();
		(*this) *= c.pow(k);
		(*this).insert((*this).begin(), l * k, 0);
		return *this;
	}
	F pow(const long long k) const { return F(*this).pow_inplace(k); }

	//Ο(NlogN)
	F sqrt(int deg = -1) const {
		auto SQRT = [&](T t) {
			int mod = T::mod();
			if(t == 0 or t == 1) return t;
			int v = (mod - 1) / 2;
			if(t.pow(v) != 1) return T(-1);
			int q = mod - 1, m = 0;
			while(~q & 1) q >>= 1, m++;
			std::mt19937 mt;
			T z = mt();
			while(z.pow(v) != mod - 1) z = mt();
			T c = z.pow(q), u = t.pow(q), r = t.pow((q + 1) / 2);
			for(; m > 1; m--) {
				T tmp = u.pow(1 << (m - 2));
				if(tmp != 1) r = r * c, u = u * c * c;
				c = c * c;
			}
			return T(std::min(r.val(), mod - r.val()));
		};
		int n = (*this).size();
		if(deg == -1) deg = n;
		if((*this)[0] == 0) {
			for(int i = 1; i < n; i++) {
				if((*this)[i] != 0) {
					if(i & 1) return F(0);
					if(deg - i / 2 <= 0) break;
					auto ret = (*this);
					ret >>= i;
					ret.resize(n - i);
					ret = ret.sqrt(deg - i / 2);
					if(ret.empty()) return F(0);
					ret <<= (i / 2);
					ret.resize(deg);
					return ret;
				}
			}
			return F(deg);
		}
		auto sqr = SQRT((*this)[0]);
		if(sqr * sqr != (*this)[0]) return F(0);
		F ret{sqr};
		T ti = T(1) / T(2);
		for(int i = 1; i < deg; i <<= 1) {
			auto u = (*this);
			u.resize(i << 1);
			ret = (ret.inv(i << 1) * u + ret) * ti;
		}
		ret.resize(deg);
		return ret;
	}

	void sparse_pow(const int n, const int d, const T c, const int k) {
		F ret(n);
		T tmp = 1;
		if(k >= 0) {
			for(int i = 0; i < n; i += d) {
				ret[i] = fact.binom(k, i / d) * tmp;
				tmp *= c;
			}
		} else {
			for(int i = 0; i < n; i += d) {
				ret[i] = fact.binom(i / d - k - 1, -k - 1) * tmp;
				tmp *= -c;
			}
		}
		(*this) = ret;
	}

	void sparse_pow_inv(const int n, const int d, const T c, const int k) { return sparse_pow(n, d, c, -k); }

	void stirling_first(int n) {
		if(!n) {
			*this = F{1};
			return;
		}
		int m = 1;
		F res(n + 1);
		res[1] = 1;
		for(int k = 30 - __builtin_clz(n); k >= 0; --k) {
			F as(m * 2 + 1), bs(m + 1);
			for(int i = 0; i <= m; i++)
				as[i] = fact.fac[i] * res[i];

			bs[m] = 1;
			for(int i = m - 1; i >= 0; i--)
				bs[i] -= bs[i + 1] * m;

			for(int i = 0; i <= m; i++)
				bs[m - i] *= fact.finv[i];

			F cs = as * bs, ds(m + 1);
			for(int i = 0; i <= m; i++)
				ds[i] = cs[m + i] * fact.finv[i];

			res *= ds;
			m <<= 1;
			if(n >> k & 1) {
				F g(n + 1);
				for(int i = 0; i <= m; i++) {
					g[i] -= res[i] * m;
					g[i + 1] += res[i];
				}
				res = g;
				m |= 1;
			}
		}
		*this = res;
	}

	void stirling_second(int n) {
		F f(n + 1), g(n + 1);
		for(int i = 0; i <= n; i++) {
			f[i] = T(i).pow(n) * fact.finv[i];
			g[i] = fact.finv[i] * (i % 2 ? -1 : 1);
		}
		f *= g;
		*this = f;
	}

	//return f(x + c)
	F taylor_shift(int c) {
		F f(*this);
		int n = this->size();
		for(int i = 0; i < n; i++) f[i] *= fact.fac[i];
		reverse(f.begin(), f.end());
		F g(n, 1);
		T mul = 1;
		for(int i = 1; i < n; i++)
			g[i] = (mul *= c) * fact.finv[i];
		f *= g;
		reverse(f.begin(), f.end());
		for(int i = 0; i < n; i++) f[i] *= fact.finv[i];
		return f;
	}
	F taylor_shift(T c) { return taylor_shift(c.val()); }

	template <class U>
	std::vector<T> multipoint_evaluation(const std::vector<U> &p) {
		using fps = formal_power_series<T, mode>;
		int m = p.size();
		int n = 1 << std::max(atcoder::internal::ceil_pow2(m), 1);
		std::vector<fps> subproducts(2 * n, F{1}), rem(2 * n);
		for(int i = n; i < n + m; i++) subproducts[i] = fps({-p[i - n], 1});
		for(int i = n - 1; i; i--) {
			int x = subproducts[i << 1].size(), y = subproducts[i << 1 | 1].size();
			subproducts[i] = subproducts[i << 1];
			subproducts[i].resize(x + y - 1);
			subproducts[i] *= subproducts[i << 1 | 1];
		}
		rem[1] = *this;
		for(int i = 1; i < n; i++) {
			rem[i << 1] = rem[i] % subproducts[i << 1];
			rem[i << 1].shrink();
			rem[i << 1 | 1] = rem[i] % subproducts[i << 1 | 1];
			rem[i << 1 | 1].shrink();
		}
		std::vector<T> res(m);
		for(int i = 0; i < m; i++) res[i] = (!rem[i + n].empty() ? rem[i + n][0] : 0);
		return res;
	}
};
#line 3 "/Users/nok0/Documents/Programming/nok0/math/bostan_mori.hpp"

template <class F>
F Berlekamp_Massey(const F &a) {
	using T = typename F::value_type;
	int n = a.size();
	F c{-1}, c2{0};
	T r2 = 1;
	int i2 = -1;
	for(int i = 0; i < n; i++) {
		T r = 0;
		int d = c.size();
		for(int j = 0; j < d; j++) r += c[j] * a[i - j];
		if(r == 0) continue;
		T coef = -r / r2;
		int d2 = c2.size();
		if(d - i >= d2 - i2) {
			for(int j = 0; j < d2; j++)
				c[j + i - i2] += c2[j] * coef;
		} else {
			F tmp(c);
			c.resize(d2 + i - i2);
			for(int j = 0; j < d2; j++) c[j + i - i2] += c2[j] * coef;
			c2 = std::move(tmp);
			i2 = i, r2 = r;
		}
	}
	return {c.begin() + 1, c.end()};
}

//return generating function of a, s.t. F(x) = P(x) / Q(x)
template <class F>
std::pair<F, F> find_generating_function(F a) {
	auto q = Berlekamp_Massey(a);
	int d = q.size();
	a.resize(d);
	q.insert(q.begin(), 1);
	for(int i = 1; i < (int)q.size(); i++) q[i] *= -1;
	a *= q;
	return {a, q};
}

//return [x^k] p(x) / q(x)
template <class T, Mode mode>
T compute_Kthterm(formal_power_series<T, mode> p, formal_power_series<T, mode> q, long long k) {
	int d = q.size();
	assert(q[0] == 1 and p.size() + 1 <= d);
	while(k) {
		auto q_minus = q;
		for(int i = 1; i < d; i += 2)
			q_minus[i] *= -1;
		p.resize(2 * d);
		q.resize(2 * d);
		p *= q_minus;
		q *= q_minus;
		for(int i = 0; i < d - 1; i++)
			p[i] = p[(i << 1) | (k & 1)];
		for(int i = 0; i < d; i++)
			q[i] = q[i << 1];
		p.resize(d - 1);
		q.resize(d);
		k >>= 1;
	}
	return p[0];
}

template <class T, Mode mode>
T compute_Kthterm(std::pair<formal_power_series<T, mode>, formal_power_series<T, mode>> f, long long k) { return compute_Kthterm(f.first, f.second, k); }
#line 3 "/Users/nok0/Documents/Programming/nok0/math/modint_iostream.hpp"

#line 5 "/Users/nok0/Documents/Programming/nok0/math/modint_iostream.hpp"
template <int m>
std::istream &std::operator>>(std::istream &is, atcoder::static_modint<m> &a) {
	long long v;
	is >> v;
	a = v;
	return is;
}
template <int m>
std::istream &std::operator>>(std::istream &is, atcoder::dynamic_modint<m> &a) {
	long long v;
	is >> v;
	a = v;
	return is;
}
template <int m>
std::ostream &std::operator<<(std::ostream &os, const atcoder::static_modint<m> &a) { return os << a.val(); }
template <int m>
std::ostream &std::operator<<(std::ostream &os, const atcoder::dynamic_modint<m> &a) { return os << a.val(); }
#line 5 "a.cpp"

using mint = atcoder::modint1000000007;
using fps = formal_power_series<mint, NAIVE>;

void main_() {
	INT(k, m);
	LL(n);
	map<pii, V<>> mp;
	REP(i, m) {
		INT(p, q, r);
		--p, --q, --r;
		mp[{p, q}].pb(r);
	}

	const int D = 1000;
	vector dp(D + 1, vector(k * k, mint(0)));
	REP(i, k) { dp[2][i] = 1; }

	fps ret(D + 1);
	REP(i, 2, D) {
		REP(j, k * k) {
			const auto &now = dp[i][j];
			auto u = j / k;
			auto v = j % k;
			for(auto nx : mp[{u, v}]) {
				dp[i + 1][v * k + nx] += now;
				if(!nx) ret[i + 1] += now;
			}
		}
	}
	print(compute_Kthterm(find_generating_function(ret), n));
}