#include <bits/stdc++.h>
// #include <x86intrin.h>
using namespace std;
using namespace numbers;

using uint = unsigned int;
template<uint _mod>
struct modular_fixed_base{
	static constexpr uint mod(){
		return _mod;
	}
	template<class T>
	static vector<modular_fixed_base> precalc_power(T base, int SZ){
		vector<modular_fixed_base> res(SZ + 1, 1);
		for(auto i = 1; i <= SZ; ++ i) res[i] = res[i - 1] * base;
		return res;
	}
	static vector<modular_fixed_base> _INV;
	static void precalc_inverse(int SZ){
		if(_INV.empty()) _INV.assign(2, 1);
		for(auto x = _INV.size(); x <= SZ; ++ x) _INV.push_back(_mod / x * -_INV[_mod % x]);
	}
	// _mod must be a prime
	static modular_fixed_base _primitive_root;
	static modular_fixed_base primitive_root(){
		if(_primitive_root) return _primitive_root;
		if(_mod == 2) return _primitive_root = 1;
		if(_mod == 998244353) return _primitive_root = 3;
		uint divs[20] = {};
		divs[0] = 2;
		int cnt = 1;
		uint x = (_mod - 1) / 2;
		while(x % 2 == 0) x /= 2;
		for(auto i = 3; 1LL * 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(auto g = 2; ; ++ g){
			bool ok = true;
			for(auto i = 0; i < cnt; ++ i){
				if((modular_fixed_base(g).power((_mod - 1) / divs[i])) == 1){
					ok = false;
					break;
				}
			}
			if(ok) return _primitive_root = g;
		}
	}
	constexpr modular_fixed_base(): data(){ }
	modular_fixed_base(const double &x){ data = normalize(llround(x)); }
	modular_fixed_base(const long double &x){ data = normalize(llround(x)); }
	template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base(const T &x){ data = normalize(x); }
	template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr> static uint normalize(const T &x){
		int sign = x >= 0 ? 1 : -1;
		uint v =  _mod <= sign * x ? sign * x % _mod : sign * x;
		if(sign == -1 && v) v = _mod - v;
		return v;
	}
	const uint &operator()() const{ return data; }
	template<class T> operator T() const{ return data; }
	modular_fixed_base &operator+=(const modular_fixed_base &otr){ if((data += otr.data) >= _mod) data -= _mod; return *this; }
	modular_fixed_base &operator-=(const modular_fixed_base &otr){ if((data += _mod - otr.data) >= _mod) data -= _mod; return *this; }
	template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base &operator+=(const T &otr){ return *this += modular_fixed_base(otr); }
	template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base &operator-=(const T &otr){ return *this -= modular_fixed_base(otr); }
	modular_fixed_base &operator++(){ return *this += 1; }
	modular_fixed_base &operator--(){ return *this += _mod - 1; }
	modular_fixed_base operator++(int){ modular_fixed_base result(*this); *this += 1; return result; }
	modular_fixed_base operator--(int){ modular_fixed_base result(*this); *this += _mod - 1; return result; }
	modular_fixed_base operator-() const{ return modular_fixed_base(_mod - data); }
	modular_fixed_base &operator*=(const modular_fixed_base &rhs){
		data = (unsigned long long)data * rhs.data % _mod;
		return *this;
	}
	template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr>
	modular_fixed_base &inplace_power(T e){
		if(!data) return *this = {};
		if(data == 1) return *this;
		if(data == mod() - 1) return e % 2 ? *this : *this = -*this;
		if(e < 0) *this = 1 / *this, e = -e;
		modular_fixed_base res = 1;
		for(; e; *this *= *this, e >>= 1) if(e & 1) res *= *this;
		return *this = res;
	}
	template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr>
	modular_fixed_base power(T e) const{
		return modular_fixed_base(*this).inplace_power(e);
	}
	modular_fixed_base &operator/=(const modular_fixed_base &otr){
		int a = otr.data, m = _mod, u = 0, v = 1;
		if(a < _INV.size()) return *this *= _INV[a];
		while(a){
			int t = m / a;
			m -= t * a; swap(a, m);
			u -= t * v; swap(u, v);
		}
		assert(m == 1);
		return *this *= u;
	}
	uint data;
};
template<uint _mod> vector<modular_fixed_base<_mod>> modular_fixed_base<_mod>::_INV;
template<uint _mod> modular_fixed_base<_mod> modular_fixed_base<_mod>::_primitive_root;
template<uint _mod> bool operator==(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return lhs.data == rhs.data; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> bool operator==(const modular_fixed_base<_mod> &lhs, T rhs){ return lhs == modular_fixed_base<_mod>(rhs); }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> bool operator==(T lhs, const modular_fixed_base<_mod> &rhs){ return modular_fixed_base<_mod>(lhs) == rhs; }
template<uint _mod> bool operator!=(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return !(lhs == rhs); }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> bool operator!=(const modular_fixed_base<_mod> &lhs, T rhs){ return !(lhs == rhs); }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> bool operator!=(T lhs, const modular_fixed_base<_mod> &rhs){ return !(lhs == rhs); }
template<uint _mod> bool operator<(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return lhs.data < rhs.data; }
template<uint _mod> bool operator>(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return lhs.data > rhs.data; }
template<uint _mod> bool operator<=(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return lhs.data <= rhs.data; }
template<uint _mod> bool operator>=(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return lhs.data >= rhs.data; }
template<uint _mod> modular_fixed_base<_mod> operator+(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return modular_fixed_base<_mod>(lhs) += rhs; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base<_mod> operator+(const modular_fixed_base<_mod> &lhs, T rhs){ return modular_fixed_base<_mod>(lhs) += rhs; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base<_mod> operator+(T lhs, const modular_fixed_base<_mod> &rhs){ return modular_fixed_base<_mod>(lhs) += rhs; }
template<uint _mod> modular_fixed_base<_mod> operator-(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return modular_fixed_base<_mod>(lhs) -= rhs; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base<_mod> operator-(const modular_fixed_base<_mod> &lhs, T rhs){ return modular_fixed_base<_mod>(lhs) -= rhs; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base<_mod> operator-(T lhs, const modular_fixed_base<_mod> &rhs){ return modular_fixed_base<_mod>(lhs) -= rhs; }
template<uint _mod> modular_fixed_base<_mod> operator*(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs){ return modular_fixed_base<_mod>(lhs) *= rhs; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base<_mod> operator*(const modular_fixed_base<_mod> &lhs, T rhs){ return modular_fixed_base<_mod>(lhs) *= rhs; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base<_mod> operator*(T lhs, const modular_fixed_base<_mod> &rhs){ return modular_fixed_base<_mod>(lhs) *= rhs; }
template<uint _mod> modular_fixed_base<_mod> operator/(const modular_fixed_base<_mod> &lhs, const modular_fixed_base<_mod> &rhs) { return modular_fixed_base<_mod>(lhs) /= rhs; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base<_mod> operator/(const modular_fixed_base<_mod> &lhs, T rhs) { return modular_fixed_base<_mod>(lhs) /= rhs; }
template<uint _mod, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_fixed_base<_mod> operator/(T lhs, const modular_fixed_base<_mod> &rhs) { return modular_fixed_base<_mod>(lhs) /= rhs; }
template<uint _mod> istream &operator>>(istream &in, modular_fixed_base<_mod> &number){
	long long x;
	in >> x;
	number.data = modular_fixed_base<_mod>::normalize(x);
	return in;
}
// #define _SHOW_FRACTION
template<uint _mod> ostream &operator<<(ostream &out, const modular_fixed_base<_mod> &number){
#if defined(LOCAL) && defined(_SHOW_FRACTION)
	out << number();
	cerr << "(";
	for(auto d = 1; ; ++ d){
		if((number * d).data <= 1000000){
			cerr << (number * d).data << "/" << d;
			break;
		}
		else if((-number * d).data <= 1000000){
			cerr << "-" << (-number * d).data << "/" << d;
			break;
		}
	}
	cerr << ")";
	return out;
#else
	return out << number();
#endif
}
#undef _SHOW_FRACTION

// const uint mod = 1e9 + 7; // 1000000007
const uint mod = (119 << 23) + 1; // 998244353
// const uint mod = 1e9 + 9; // 1000000009
using modular = modular_fixed_base<mod>;

template<class T>
struct combinatorics{
	// O(n)
	static vector<T> precalc_fact(int n){
		vector<T> f(n + 1, 1);
		for(auto i = 1; i <= n; ++ i) f[i] = f[i - 1] * i;
		return f;
	}
	// O(n * m)
	static vector<vector<T>> precalc_C(int n, int m){
		vector<vector<T>> c(n + 1, vector<T>(m + 1));
		for(auto i = 0; i <= n; ++ i) for(auto j = 0; j <= min(i, m); ++ j) c[i][j] = i && j ? c[i - 1][j - 1] + c[i - 1][j] : T(1);
		return c;
	}
	int SZ = 0;
	vector<T> inv, fact, invfact;
	combinatorics(){ }
	// O(SZ)
	combinatorics(int SZ): SZ(SZ), inv(SZ + 1, 1), fact(SZ + 1, 1), invfact(SZ + 1, 1){
		for(auto i = 1; i <= SZ; ++ i) fact[i] = fact[i - 1] * i;
		invfact[SZ] = 1 / fact[SZ];
		for(auto i = SZ - 1; i >= 0; -- i){
			invfact[i] = invfact[i + 1] * (i + 1);
			inv[i + 1] = invfact[i + 1] * fact[i];
		}
	}
	// O(1)
	T C(int n, int k) const{
		return n < 0 ? C(-n + k - 1, k) * (k & 1 ? -1 : 1) : n < k || k < 0 ? T() : fact[n] * invfact[k] * invfact[n - k];
	}
	// O(1)
	T P(int n, int k) const{
		return n < k ? T() : fact[n] * invfact[n - k];
	}
	// O(1)
	T H(int n, int k) const{
		return C(n + k - 1, k);
	}
	// O(min(k, n - k))
	T naive_C(long long n, long long k) const{
		if(n < k) return 0;
		T res = 1;
		k = min(k, n - k);
		for(auto i = n; i > n - k; -- i) res *= i;
		return res * invfact[k];
	}
	// O(k)
	T naive_P(long long n, int k) const{
		if(n < k) return 0;
		T res = 1;
		for(auto i = n; i > n - k; -- i) res *= i;
		return res;
	}
	// O(k)
	T naive_H(long long n, int k) const{
		return naive_C(n + k - 1, k);
	}
	// O(1)
	bool parity_C(long long n, long long k) const{
		return n < k ? false : (n & k) == k;
	}
	// Number of ways to place n '('s and k ')'s starting with m copies of '(' such that in each prefix, number of '(' is equal or greater than ')'
	// Catalan(n, n, 0): n-th catalan number
	// Catalan(s, s+k-1, k-1): sum of products of k catalan numbers where the index of product sums up to s.
	// O(1)
	T Catalan(int n, int k, int m = 0) const{
		assert(0 <= min({n, k, m}));
		return k <= m ? C(n + k, n) : k <= n + m ? C(n + k, n) - C(n + k, k - m - 1) : T();
	}
};

// T must be of modular type
// mod must be a prime
// Requires modular
template<class T>
struct number_theoric_transform_wrapper{
	// i \in [2^k, 2^{k+1}) holds w_{2^k+1}^{i-2^k}
	static vector<T> root, buffer1, buffer2;
	static void adjust_root(int n){
		if(root.empty()) root = {1, 1};
		for(auto k = (int)root.size(); k < n; k <<= 1){
			root.resize(n, 1);
			T w = T::primitive_root().power((T::mod() - 1) / (k << 1));
			for(auto i = k; i < k << 1; ++ i) root[i] = i & 1 ? root[i >> 1] * w : root[i >> 1];
		}
	}
	// n must be a power of two
	// p must have next n memories allocated
	// O(n * log(n))
	static void transform(int n, T *p, bool invert = false){
		assert(n && __builtin_popcount(n) == 1 && (T::mod() - 1) % n == 0);
		for(auto i = 1, j = 0; i < n; ++ i){
			int bit = n >> 1;
			for(; j & bit; bit >>= 1) j ^= bit;
			j ^= bit;
			if(i < j) swap(p[i], p[j]);
		}
		adjust_root(n);
		for(auto len = 1; len < n; len <<= 1) for(auto i = 0; i < n; i += len << 1) for(auto j = 0; j < len; ++ j){
			T x = p[i + j], y = p[len + i + j] * root[len + j];
			p[i + j] = x + y, p[len + i + j] = x - y;
		}
		if(invert){
			reverse(p + 1, p + n);
			T inv_n = T(1) / n;
			for(auto i = 0; i < n; ++ i) p[i] *= inv_n;
		}
	}
	static void transform(vector<T> &p, bool invert = false){
		transform((int)p.size(), p.data(), invert);
	}
	// Double the length of the ntt array
	// n must be a power of two
	// p must have next 2n memories allocated
	// O(n * log(n))
	static void double_up(int n, T *p){
		assert(n && __builtin_popcount(n) == 1 && (T().mod() - 1) % (n << 1) == 0);
		buffer1.resize(n << 1);
		for(auto i = 0; i < n; ++ i) buffer1[i << 1] = p[i];
		transform(n, p, true);
		adjust_root(n << 1);
		for(auto i = 0; i < n; ++ i) p[i] *= root[n | i];
		transform(n, p);
		for(auto i = 0; i < n; ++ i) buffer1[i << 1 | 1] = p[i];
		copy(buffer1.begin(), buffer1.begin() + 2 * n, p);
	}
	static void double_up(vector<T> &p){
		int n = (int)p.size();
		p.resize(n << 1);
		double_up(n, p.data());
	}
	// O(n * m)
	static vector<T> convolute_naive(const vector<T> &p, const vector<T> &q){
		vector<T> res(max((int)p.size() + (int)q.size() - 1, 0));
		for(auto i = 0; i < (int)p.size(); ++ i) for(auto j = 0; j < (int)q.size(); ++ j) res[i + j] += p[i] * q[j];
		return res;
	}
	// O((n + m) * log(n + m))
	static vector<T> convolute(const vector<T> &p, const vector<T> &q){
		if(min(p.size(), q.size()) < 55) return convolute_naive(p, q);
		int m = (int)p.size() + (int)q.size() - 1, n = 1 << __lg(m) + 1;
		buffer1.assign(n, 0);
		copy(p.begin(), p.end(), buffer1.begin());
		transform(buffer1);
		buffer2.assign(n, 0);
		copy(q.begin(), q.end(), buffer2.begin());
		transform(buffer2);
		for(auto i = 0; i < n; ++ i) buffer1[i] *= buffer2[i];
		transform(buffer1, true);
		return vector<T>(buffer1.begin(), buffer1.begin() + m);
	}
	// O((n + m) * log(n + m))
	static vector<T> arbitrarily_convolute(const vector<T> &p, const vector<T> &q){
		using modular0 = modular_fixed_base<1045430273>;
		using modular1 = modular_fixed_base<1051721729>;
		using modular2 = modular_fixed_base<1053818881>;
		using ntt0 = number_theoric_transform_wrapper<modular0>;
		using ntt1 = number_theoric_transform_wrapper<modular1>;
		using ntt2 = number_theoric_transform_wrapper<modular2>;
		vector<modular0> p0((int)p.size()), q0((int)q.size());
		for(auto i = 0; i < (int)p.size(); ++ i) p0[i] = p[i].data;
		for(auto i = 0; i < (int)q.size(); ++ i) q0[i] = q[i].data;
		auto xy0 = ntt0::convolute(p0, q0);
		vector<modular1> p1((int)p.size()), q1((int)q.size());
		for(auto i = 0; i < (int)p.size(); ++ i) p1[i] = p[i].data;
		for(auto i = 0; i < (int)q.size(); ++ i) q1[i] = q[i].data;
		auto xy1 = ntt1::convolute(p1, q1);
		vector<modular2> p2((int)p.size()), q2((int)q.size());
		for(auto i = 0; i < (int)p.size(); ++ i) p2[i] = p[i].data;
		for(auto i = 0; i < (int)q.size(); ++ i) q2[i] = q[i].data;
		auto xy2 = ntt2::convolute(p2, q2);
		static const modular1 r01 = 1 / modular1(modular0::mod());
		static const modular2 r02 = 1 / modular2(modular0::mod());
		static const modular2 r12 = 1 / modular2(modular1::mod());
		static const modular2 r02r12 = r02 * r12;
		static const T w1 = modular0::mod();
		static const T w2 = w1 * modular1::mod();
		int n = (int)p.size() + (int)q.size() - 1;
		vector<T> res(n);
		for(auto i = 0; i < n; ++ i){
			using ull = unsigned long long;
			ull a = xy0[i].data;
			ull b = (xy1[i].data + modular1::mod() - a) * r01.data % modular1::mod();
			ull c = ((xy2[i].data + modular2::mod() - a) * r02r12.data + (modular2::mod() - b) * r12.data) % modular2::mod();
			res[i] = xy0[i].data + w1 * b + w2 * c;
		}
		return res;
	}
};
template<class T> vector<T> number_theoric_transform_wrapper<T>::root;
template<class T> vector<T> number_theoric_transform_wrapper<T>::buffer1;
template<class T> vector<T> number_theoric_transform_wrapper<T>::buffer2;

using ntt = number_theoric_transform_wrapper<modular>;

// T must support *=, *, +=, -=, and -
// T{} should return the additive identity
// Requires modular and number_theoric_transform
template<class T, class FFT>
struct power_series_base: vector<T>{
#define data (*this)
	template<class ...Args>
	power_series_base(Args... args): vector<T>(args...){}
	power_series_base(initializer_list<T> init): vector<T>(init){}
	int degree() const{
		return data.empty() ? numeric_limits<int>::max() : data.size() - 1;
	}
	// Returns \sum_{i=0}^{n-1} a_i/i! * X^i
	static power_series_base EGF(vector<T> a){
		int n = (int)a.size();
		T fact = 1;
		for(auto x = 2; x < n; ++ x) fact *= x;
		fact = 1 / fact;
		for(auto i = n - 1; i >= 0; -- i) a[i] *= fact, fact *= i;
		return power_series_base(a);
	}
	// Returns exp(coef * X).prefix(n) = \sum_{i=0}^{n-1} coef^i/i! * X^i
	static power_series_base EGF(int n, T coef = 1){
		vector<T> a(n, 1);
		for(auto i = 1; i < n; ++ i) a[i] = a[i - 1] * coef;
		return EGF(a);
	}
	vector<T> EGF_to_seq() const{
		int n = (int)data.size();
		vector<T> seq(n);
		T fact = 1;
		for(auto i = 0; i < n; ++ i){
			seq[i] = data[i] * fact;
			fact *= i + 1;
		}
		return seq;
	}
	power_series_base &reduce(){
		while(!data.empty() && !data.back()) data.pop_back();
		return *this;
	}
	power_series_base reduced() const{
		return power_series_base(*this).reduce();
	}
	friend ostream &operator<<(ostream &out, const power_series_base &p){
		if(p.empty()) return out << "{}";
		else{
			out << "{";
			for(auto i = 0; i < (int)p.size(); ++ i){
				out << p[i];
				i + 1 < (int)p.size() ? out << ", " : out << "}";
			}
			return out;
		}
	}
	power_series_base &inplace_take(int n){
		data.erase(data.begin() + min((int)data.size(), n), data.end());
		data.resize(n, T{});
		return *this;
	}
	power_series_base take(int n) const{
		return power_series_base(*this).inplace_take(n);
	}
	power_series_base &inplace_drop(int n){
		data.erase(data.begin(), data.begin() + min((int)data.size(), n));
		return *this;
	}
	power_series_base drop(int n) const{
		return power_series_base(*this).inplace_drop(n);
	}
	power_series_base &inplace_reverse(int n){
		data.resize(max(n, (int)data.size()));
		std::reverse(data.begin(), data.begin() + n);
		return *this;
	}
	power_series_base reverse(int n) const{
		return power_series_base(*this).inplace_reverse(n);
	}
	T evaluate(T x) const{
		T res = {};
		for(auto i = (int)data.size() - 1; i >= 0; -- i) res = res * x + data[i];
		return res;
	}
	power_series_base operator*(const power_series_base &p) const{
		return FFT::convolute(data, p);
	}
	power_series_base &operator*=(const power_series_base &p){
		return *this = *this * p;
	}
	template<class U>
	power_series_base &operator*=(U x){
		for(auto &c: data) c *= x;
		return *this;
	}
	template<class U>
	power_series_base operator*(U x) const{
		return power_series_base(*this) *= x;
	}
	template<class U>
	friend power_series_base operator*(U x, power_series_base p){
		for(auto &c: p) c = x * c;
		return p;
	}
	power_series_base &operator+=(const power_series_base &p){
		data.resize(max(data.size(), p.size()));
		for(auto i = 0; i < (int)p.size(); ++ i) data[i] += p[i];
		return *this;
	}
	power_series_base operator+(const power_series_base &p) const{
		return power_series_base(*this) += p;
	}
	template<class U>
	power_series_base &operator+=(const U &x){
		if(data.empty()) data.emplace_back();
		data[0] += x;
		return *this;
	}
	template<class U>
	power_series_base operator+(const U &x) const{
		return power_series_base(*this) += x;
	}
	template<class U>
	friend power_series_base operator+(const U &x, const power_series_base &p){
		return p + x;
	}
	power_series_base &operator-=(const power_series_base &p){
		data.resize(max(data.size(), p.size()));
		for(auto i = 0; i < (int)p.size(); ++ i) data[i] -= p[i];
		return *this;
	}
	power_series_base operator-(const power_series_base &p) const{
		return power_series_base(*this) -= p;
	}
	template<class U>
	power_series_base &operator-=(const U &x){
		if(data.empty()) data.emplace_back();
		data[0] -= x;
		return *this;
	}
	template<class U>
	power_series_base operator-(const U &x) const{
		return power_series_base(*this) -= x;
	}
	template<class U>
	friend power_series_base operator-(const U &x, const power_series_base &p){
		return -p + x;
	}
	power_series_base operator-() const{
		power_series_base res = *this;
		for(auto i = 0; i < data.size(); ++ i) res[i] = T{} - res[i];
		return res;
	}
	power_series_base &operator++(){
		if(data.empty()) data.push_back(1);
		else ++ data[0];
		return *this;
	}
	power_series_base &operator--(){
		if(data.empty()) data.push_back(-1);
		else -- data[0];
		return *this;
	}
	power_series_base operator++(int){
		power_series_base result(*this);
		if(data.empty()) data.push_back(1);
		else ++ data[0];
		return result;
	}
	power_series_base operator--(int){
		power_series_base result(*this);
		if(data.empty()) data.push_back(-1);
		else -- data[0];
		return result;
	}
	power_series_base &clear_range(int l, int r){
		assert(0 <= l && l <= r && r <= data.size());
		for(auto i = l; i < r; ++ i) data[i] = 0;
		return *this;
	}
	power_series_base &transform(bool invert = false){
		FFT::transform(data, invert);
		return *this;
	}
	power_series_base transformed(bool invert = false) const{
		auto res = *this;
		res.transform(invert);
		return res;
	}
	power_series_base &inplace_dot_product(const power_series_base &p){
		for(auto i = 0; i < min(data.size(), p.size()); ++ i) data[i] *= p[i];
		return *this;
	}
	power_series_base dot_product(const power_series_base &p) const{
		return power_series_base(*this).inplace_power_series_product(p);
	}
	power_series_base &inverse_doubled_up(power_series_base &f, const power_series_base &freq) const{
		assert((f.size() & -f.size()) == f.size());
		int s = f.size();
		power_series_base buffer = take(s << 1);
		buffer.transform();
		buffer.inplace_dot_product(freq);
		buffer.transform(true);
		buffer.clear_range(0, s);
		buffer.transform();
		buffer.inplace_dot_product(freq);
		buffer.transform(true);
		f.resize(s << 1);
		return f -= buffer.clear_range(0, s);
	}
	power_series_base &inverse_doubled_up(power_series_base &f) const{
		assert((f.size() & -f.size()) == f.size());
		return inverse_doubled_up(f, f.take(f.size() << 1).transformed());
	}
	// Returns the first n terms of the inverse series
	// O(n * log(n))
	power_series_base inverse(int n) const{
		assert(!data.empty() && data[0]);
		auto inv = 1 / data[0];
		power_series_base res{inv};
		for(auto s = 1; s < n; s <<= 1) inverse_doubled_up(res);
		res.resize(n);
		return res;
	}
	// Returns the first n terms of the inverse series
	// O(n * log(n))
	power_series_base &inplace_inverse(int n){
		return *this = this->inverse(n);
	}
	// O(n * log(n))
	power_series_base &inplace_power_series_division(power_series_base p, int n){
		int i = 0;
		while(i < min(data.size(), p.size()) && !data[i] && !p[i]) ++ i;
		data.erase(data.begin(), data.begin() + i);
		p.erase(p.begin(), p.begin() + i);
		(*this *= p.inverse(n)).resize(n);
		return *this;
	}
	// O(n * log(n))
	power_series_base power_series_division(const power_series_base &p, int n){
		return power_series_base(*this).inplace_power_series_division(p, n);
	}
	// Euclidean division
	// O(n * log(n))
	power_series_base &operator/=(const power_series_base &p){
		assert(!p.empty() && p.back() != T{0});
		reduce();
		if(data.size() < p.size()){
			data.clear();
			return *this;
		}
		if((int)p.size() - count(p.begin(), p.end(), 0) <= 100){
			T inv = 1 / p.back();
			static vector<int> indices;
			for(auto i = 0; i < (int)p.size() - 1; ++ i) if(p[i]) indices.push_back(i);
			power_series_base res((int)data.size() - (int)p.size() + 1);
			for(auto i = (int)data.size() - 1; i >= (int)p.size() - 1; -- i) if(data[i]){
				T x = data[i] * inv;
				res[i - (int)p.size() + 1] = x;
				for(auto j: indices) data[i - ((int)p.size() - 1 - j)] -= x * p[j];
			}
			indices.clear();
			return *this = res;
		}
		power_series_base b;
		int n = data.size() - p.size() + 1;
		b.assign(n, {});
		copy(p.rbegin(), p.rbegin() + min(p.size(), b.size()), b.begin());
		std::reverse(data.begin(), data.end());
		data = FFT::convolute(data, b.inverse(n));
		data.erase(data.begin() + n, data.end());
		std::reverse(data.begin(), data.end());
		return *this;
	}
	power_series_base operator/(const power_series_base &p) const{
		return power_series_base(*this) /= p;
	}
	template<class U>
	power_series_base &operator/=(U x){
		assert(x);
		T inv_x = T(1) / x;
		for(auto &c: data) c *= inv_x;
		return *this;
	}
	template<class U>
	power_series_base operator/(U x) const{
		return power_series_base(*this) /= x;
	}
	pair<power_series_base, power_series_base> divrem(const power_series_base &p) const{
		auto q = *this / p, r = *this - q * p;
		while(!r.empty() && r.back() == 0) r.pop_back();
		return {q, r};
	}
	power_series_base &operator%=(const power_series_base &p){
		assert(!p.empty() && p.back() != T{0});
		reduce();
		if(data.size() < p.size()) return *this;
		if((int)p.size() - count(p.begin(), p.end(), 0) <= 100){
			T inv = 1 / p.back();
			static vector<int> indices;
			for(auto i = 0; i < (int)p.size() - 1; ++ i) if(p[i]) indices.push_back(i);
			for(auto i = (int)data.size() - 1; i >= (int)p.size() - 1; -- i) if(data[i]){
				T x = data[i] * inv;
				data[i] = 0;
				for(auto j: indices) data[i - ((int)p.size() - 1 - j)] -= x * p[j];
			}
			indices.clear();
			return reduce();
		}
		return *this = this->divrem(p).second;
	}
	power_series_base operator%(const power_series_base &p) const{
		return power_series_base(*this) %= p;
	}
	power_series_base &inplace_derivative(){
		if(!data.empty()){
			for(auto i = 0; i < data.size(); ++ i) data[i] *= i;
			data.erase(data.begin());
		}
		return *this;
	}
	// p'
	power_series_base derivative() const{
		return power_series_base(*this).inplace_derivative();
	}
	power_series_base &inplace_derivative_shift(){
		for(auto i = 0; i < data.size(); ++ i) data[i] *= i;
		return *this;
	}
	// xP'
	power_series_base derivative_shift() const{
		return power_series_base(*this).inplace_derivative_shift();
	}
	power_series_base &inplace_antiderivative(){
		T::precalc_inverse(data.size());
		data.push_back(0);
		for(auto i = (int)data.size() - 1; i >= 1; -- i) data[i] = data[i - 1] / i;
		data[0] = 0;
		return *this;
	}
	// Integral(P)
	power_series_base antiderivative() const{
		return power_series_base(*this).inplace_antiderivative();
	}
	power_series_base &inplace_shifted_antiderivative(){
		T::precalc_inverse(data.size());
		if(!data.empty()) data[0] = 0;
		for(auto i = 1; i < data.size(); ++ i) data[i] /= i;
		return *this;
	}
	// Integral(P/x)
	power_series_base shifted_antiderivative() const{
		return power_series_base(*this).inplace_shifted_antiderivative();
	}
	// O(n * log(n))
	power_series_base &inplace_log(int n){
		assert(!data.empty() && data[0] == 1);
		if(!n){
			data.clear();
			return *this;
		}
		(*this = derivative() * inverse(n)).resize(n - 1);
		inplace_antiderivative();
		return *this;
	}
	// O(n * log(n))
	power_series_base log(int n) const{
		return power_series_base(*this).inplace_log(n);
	}
	// O(n * log(n))
	power_series_base exp(int n) const{
		assert(data.empty() || data[0] == 0);
		power_series_base f{1}, g{1};
		for(auto s = 1; s < n; s <<= 1){
			power_series_base f2 = f.take(s << 1).transform();
			power_series_base g2 = g.take(s << 1).transform();
			power_series_base dt = take(s).inplace_derivative_shift();
			power_series_base w = dt;
			w.transform();
			for(auto i = 0; i < s; ++ i) w[i] *= f2[i << 1];
			w.transform(true);
			w -= f.derivative_shift();
			w.resize(s << 1);
			w.transform();
			w.inplace_dot_product(g2);
			w.transform(true);
			w.resize(s);
			w.insert(w.begin(), s, 0);
			w -= dt;
			power_series_base z = take(s << 1);
			z += w.inplace_shifted_antiderivative();
			z.transform();
			z.inplace_dot_product(f2);
			z.transform(true);
			f.resize(s << 1);
			f += z.clear_range(0, s);
			if(s << 1 < n) f.inverse_doubled_up(g, g2);
		}
		f.resize(n);
		return f;
	}
	// O(n * log(n))
	power_series_base &inplace_exp(int n){
		return *this = this->exp(n);
	}
	// O(n * log(n))
	template<class U>
	power_series_base &inplace_power(U e, int n){
		data.resize(n);
		if(e == 0 || n == 0){
			if(n) data[0] = 1;
			return *this;
		}
		if(e < 0) return inplace_inverse(n).inplace_power(-e, n);
		if(all_of(data.begin(), data.end(), [&](auto x){ return x == 0; })) return *this;
		int pivot = find_if(data.begin(), data.end(), [&](auto x){ return x; }) - data.begin();
		if(pivot && e >= (n + pivot - 1) / pivot){
			fill(data.begin(), data.end(), 0);
			return *this;
		}
		data.erase(data.begin(), data.begin() + pivot);
		n -= pivot * e;
		T pivot_c = data[0].power(e);
		((*this /= data[0]).inplace_log(n) *= e).inplace_exp(n);
		data.insert(data.begin(), pivot * e, 0);
		return *this *= pivot_c;
	}
	// O(n * log(n))
	template<class U>
	power_series_base power(U e, int n) const{
		return power_series_base(*this).inplace_power(e, n);
	}
	static power_series_base multiply_all(const vector<power_series_base> &a){
		if(a.empty()) return {1};
		auto solve = [&](auto self, int l, int r)->power_series_base{
			if(r - l == 1) return a[l];
			int m = l + (r - l >> 1);
			return self(self, l, m) * self(self, m, r);
		};
		return solve(solve, 0, (int)a.size());
	}
#undef data
};

using power_series = power_series_base<modular, ntt>;

int main(){
	cin.tie(0)->sync_with_stdio(0);
	cin.exceptions(ios::badbit | ios::failbit);
	combinatorics<modular> C(400000);
	int n, m;
	cin >> n >> m;
	vector<int> a(n);
	copy_n(istream_iterator<int>(cin), n, a.begin());
	int one_cnt = a[n - 1];
	for(auto i = n - 1; i >= 1; -- i){
		a[i] -= a[i - 1];
	}
	vector<int> posi(n), nega(n);
	for(auto i = 1; i < n; ++ i){
		if(a[i] < 0){
			nega[i] = -a[i];
		}
		else{
			posi[i] = a[i];
		}
	}
	if(a[0] < accumulate(nega.begin(), nega.end(), 0)){
		cout << "0\n";
		return 0;
	}
	int psum = accumulate(posi.begin(), posi.end(), 0);
	int nsum = accumulate(nega.begin(), nega.end(), 0);
	int c = a[0] - accumulate(nega.begin(), nega.end(), 0);
	vector<power_series> p(n - 1, vector<modular>(c + 1));
	for(auto i = 1; i < n; ++ i){
		for(auto x = 0; x <= c; ++ x){
			p[i - 1][x] = C.invfact[posi[i] + x] * C.invfact[nega[i] + x];
		}
	}
	auto prod = power_series::multiply_all(p);
	modular res = 0;
	for(auto x = 0; x <= c; ++ x){
		if(c + x + psum + nsum <= m){
			res += prod[x] * C.invfact[c - x] * C.invfact[m - c - x - psum - nsum];
		}
	}
	cout << res * C.fact[m] << "\n";
	return 0;
}

/*

*/

////////////////////////////////////////////////////////////////////////////////////////
//                                                                                    //
//                                   Coded by Aeren                                   //
//                                                                                    //
////////////////////////////////////////////////////////////////////////////////////////