// QCFium 法
//#pragma GCC target("avx2") // yukicoder と codechef では消す
#pragma GCC optimize("O3") // たまにバグる
#pragma GCC optimize("unroll-loops")


#ifndef HIDDEN_IN_VS // 折りたたみ用

// 警告の抑制
#define _CRT_SECURE_NO_WARNINGS

// ライブラリの読み込み
#include <bits/stdc++.h>
using namespace std;

// 型名の短縮
using ll = long long; using ull = unsigned long long; // -2^63 ～ 2^63 = 9e18（int は -2^31 ～ 2^31 = 2e9）
using pii = pair<int, int>;	using pll = pair<ll, ll>;	using pil = pair<int, ll>;	using pli = pair<ll, int>;
using vi = vector<int>;		using vvi = vector<vi>;		using vvvi = vector<vvi>;	using vvvvi = vector<vvvi>;
using vl = vector<ll>;		using vvl = vector<vl>;		using vvvl = vector<vvl>;	using vvvvl = vector<vvvl>;
using vb = vector<bool>;	using vvb = vector<vb>;		using vvvb = vector<vvb>;
using vc = vector<char>;	using vvc = vector<vc>;		using vvvc = vector<vvc>;
using vd = vector<double>;	using vvd = vector<vd>;		using vvvd = vector<vvd>;
template <class T> using priority_queue_rev = priority_queue<T, vector<T>, greater<T>>;
using Graph = vvi;

// 定数の定義
const double PI = acos(-1);
int DX[4] = { 1, 0, -1, 0 }; // 4 近傍（下，右，上，左）
int DY[4] = { 0, 1, 0, -1 };
int INF = 1001001001; ll INFL = 4004004003094073385LL; // (int)INFL = INF, (int)(-INFL) = -INF;

// 入出力高速化
struct fast_io { fast_io() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(18); } } fastIOtmp;

// 汎用マクロの定義
#define all(a) (a).begin(), (a).end()
#define sz(x) ((int)(x).size())
#define lbpos(a, x) (int)distance((a).begin(), std::lower_bound(all(a), (x)))
#define ubpos(a, x) (int)distance((a).begin(), std::upper_bound(all(a), (x)))
#define Yes(b) {cout << ((b) ? "Yes\n" : "No\n");}
#define rep(i, n) for(int i = 0, i##_len = int(n); i < i##_len; ++i) // 0 から n-1 まで昇順
#define repi(i, s, t) for(int i = int(s), i##_end = int(t); i <= i##_end; ++i) // s から t まで昇順
#define repir(i, s, t) for(int i = int(s), i##_end = int(t); i >= i##_end; --i) // s から t まで降順
#define repe(v, a) for(const auto& v : (a)) // a の全要素（変更不可能）
#define repea(v, a) for(auto& v : (a)) // a の全要素（変更可能）
#define repb(set, d) for(int set = 0, set##_ub = 1 << int(d); set < set##_ub; ++set) // d ビット全探索（昇順）
#define repis(i, set) for(int i = lsb(set), bset##i = set; i < 32; bset##i -= 1 << i, i = lsb(bset##i)) // set の全要素（昇順）
#define repp(a) sort(all(a)); for(bool a##_perm = true; a##_perm; a##_perm = next_permutation(all(a))) // a の順列全て（昇順）
#define uniq(a) {sort(all(a)); (a).erase(unique(all(a)), (a).end());} // 重複除去
#define EXIT(a) {cout << (a) << endl; exit(0);} // 強制終了
#define inQ(x, y, u, l, d, r) ((u) <= (x) && (l) <= (y) && (x) < (d) && (y) < (r)) // 半開矩形内判定

// 汎用関数の定義
template <class T> inline ll powi(T n, int k) { ll v = 1; rep(i, k) v *= n; return v; }
template <class T> inline bool chmax(T& M, const T& x) { if (M < x) { M = x; return true; } return false; } // 最大値を更新（更新されたら true を返す）
template <class T> inline bool chmin(T& m, const T& x) { if (m > x) { m = x; return true; } return false; } // 最小値を更新（更新されたら true を返す）
template <class T> inline int getb(T set, int i) { return (set >> i) & T(1); }
template <class T> inline T smod(T n, T m) { n %= m; if (n < 0) n += m; return n; } // 非負mod

// 演算子オーバーロード
template <class T, class U> inline istream& operator>>(istream& is, pair<T, U>& p) { is >> p.first >> p.second; return is; }
template <class T> inline istream& operator>>(istream& is, vector<T>& v) { repea(x, v) is >> x; return is; }
template <class T> inline vector<T>& operator--(vector<T>& v) { repea(x, v) --x; return v; }
template <class T> inline vector<T>& operator++(vector<T>& v) { repea(x, v) ++x; return v; }

#endif // 折りたたみ用


#if __has_include(<atcoder/all>)
#include <atcoder/all>
using namespace atcoder;

#ifdef _MSC_VER
#include "localACL.hpp"
#endif

using mint = modint998244353;
//using mint = static_modint<(int)1e9+7>;
//using mint = modint; // mint::set_mod(m);

using vm = vector<mint>; using vvm = vector<vm>; using vvvm = vector<vvm>; using vvvvm = vector<vvvm>; using pim = pair<int, mint>;
#endif


#ifdef _MSC_VER // 手元環境（Visual Studio）
#include "local.hpp"
#else // 提出用（gcc）
int mute_dump = 0;
int frac_print = 0;
#if __has_include(<atcoder/all>)
namespace atcoder {
	inline istream& operator>>(istream& is, mint& x) { ll x_; is >> x_; x = x_; return is; }
	inline ostream& operator<<(ostream& os, const mint& x) { os << x.val(); return os; }
}
#endif
inline int popcount(int n) { return __builtin_popcount(n); }
inline int popcount(ll n) { return __builtin_popcountll(n); }
inline int lsb(int n) { return n != 0 ? __builtin_ctz(n) : 32; }
inline int lsb(ll n) { return n != 0 ? __builtin_ctzll(n) : 64; }
inline int msb(int n) { return n != 0 ? (31 - __builtin_clz(n)) : -1; }
inline int msb(ll n) { return n != 0 ? (63 - __builtin_clzll(n)) : -1; }
#define dump(...)
#define dumpel(v)
#define dump_math(v)
#define input_from_file(f)
#define output_to_file(f)
#define Assert(b) { if (!(b)) { vc MLE(1<<30); rep(i,9)cout<<MLE[i]; exit(0); } } // RE の代わりに MLE を出す
#endif


// i=n, j=m に対する愚直解を返す．
mint naive_sub(int n, int m) {
	mint res = 0;

	vm a(n);
	a[m] = 1;

	// a の全ての部分列 b について
	repb(set, n) {
		vm b;
		repis(i, set) b.push_back(a[i]);
		int m = sz(b);
		
		// b の全ての連続部分列 c について
		rep(l, m) repi(r, l + 1, m) {
			// c の総和を求め答えに加える．
			repi(j, l, r - 1) res += b[j];
		}
	}

	return res;
}


// (i,j)∈[0..n)×[0..m) に対する愚直解を返す．
vvm naive() {
	int n = 10;

	vvm tbl(n);
	rep(i, n) {
		tbl[i].resize(n);
		rep(j, i) {
			tbl[i][j] = naive_sub(i, j);
		}
	}


#ifdef _MSC_VER
	// 埋め込み用
	string eb;
	eb += "vvm tbl = {\n";
	rep(i, sz(tbl)) {
		eb += "{";
		rep(j, sz(tbl[i])) eb += to_string(tbl[i][j].val()) + ",";
		eb.pop_back();
		eb += "},\n";
	}
	eb.pop_back(); eb.pop_back();
	eb += "};\n\n";
	cout << eb;
#endif

	return tbl;
}


//【行列】
template <class T>
struct Matrix {
	int n, m; // 行列のサイズ（n 行 m 列）
	vector<vector<T>> v; // 行列の成分

	// n×m 零行列で初期化する．
	Matrix(int n, int m) : n(n), m(m), v(n, vector<T>(m)) {}

	// n×n 単位行列で初期化する．
	Matrix(int n) : n(n), m(n), v(n, vector<T>(n)) { rep(i, n) v[i][i] = T(1); }

	// 二次元配列 a[0..n)[0..m) の要素で初期化する．
	Matrix(const vector<vector<T>>& a) : n(sz(a)), m(sz(a[0])), v(a) {}
	Matrix() : n(0), m(0) {}

	// 代入
	Matrix(const Matrix&) = default;
	Matrix& operator=(const Matrix&) = default;

	// アクセス
	inline vector<T> const& operator[](int i) const { return v[i]; }
	inline vector<T>& operator[](int i) {return v[i];}

	// 入力
	friend istream& operator>>(istream& is, Matrix& a) {
		rep(i, a.n) rep(j, a.m) is >> a.v[i][j];
		return is;
	}

	// 行の追加
	void push_back(const vector<T>& a) {
		Assert(sz(a) == m);
		v.push_back(a);
		n++;
	}

	// 行の削除
	void pop_back() {
		Assert(n > 0);
		v.pop_back();
		n--;
	}

	// サイズ変更
	void resize(int n_) {
		v.resize(n_);
		n = n_;
	}

	void resize(int n_, int m_) {
		n = n_;
		m = m_;

		v.resize(n);
		rep(i, n) v[i].resize(m);
	}

	// 空か
	bool empty() const { return min(n, m) == 0; }

	// 比較
	bool operator==(const Matrix& b) const { return n == b.n && m == b.m && v == b.v; }
	bool operator!=(const Matrix& b) const { return !(*this == b); }

	// 加算，減算，スカラー倍
	Matrix& operator+=(const Matrix& b) {
		rep(i, n) rep(j, m) v[i][j] += b[i][j];
		return *this;
	}
	Matrix& operator-=(const Matrix& b) {
		rep(i, n) rep(j, m) v[i][j] -= b[i][j];
		return *this;
	}
	Matrix& operator*=(const T& c) {
		rep(i, n) rep(j, m) v[i][j] *= c;
		return *this;
	}
	Matrix operator+(const Matrix& b) const { return Matrix(*this) += b; }
	Matrix operator-(const Matrix& b) const { return Matrix(*this) -= b; }
	Matrix operator*(const T& c) const { return Matrix(*this) *= c; }
	friend Matrix operator*(const T& c, const Matrix<T>& a) { return a * c; }
	Matrix operator-() const { return Matrix(*this) *= T(-1); }

	// 行列ベクトル積 : O(m n)
	vector<T> operator*(const vector<T>& x) const {
		vector<T> y(n);
		rep(i, n) rep(j, m)	y[i] += v[i][j] * x[j];
		return y;
	}

	// ベクトル行列積 : O(m n)
	friend vector<T> operator*(const vector<T>& x, const Matrix& a) {
		vector<T> y(a.m);
		rep(i, a.n) rep(j, a.m) y[j] += x[i] * a[i][j];
		return y;
	}

	// 積：O(n^3)
	Matrix operator*(const Matrix& b) const {
		Matrix res(n, b.m);
		rep(i, res.n) rep(k, m) rep(j, res.m) res[i][j] += v[i][k] * b[k][j];
		return res;
	}
	Matrix& operator*=(const Matrix& b) { *this = *this * b; return *this; }

	// 累乗：O(n^3 log d)
	Matrix pow(ll d) const {
		Matrix res(n), pow2 = *this;
		while (d > 0) {
			if (d & 1) res *= pow2;
			pow2 *= pow2;
			d >>= 1;
		}
		return res;
	}

#ifdef _MSC_VER
	friend ostream& operator<<(ostream& os, const Matrix& a) {
		rep(i, a.n) {
			os << "[";
			rep(j, a.m) os << a[i][j] << " ]"[j == a.m - 1];
			if (i < a.n - 1) os << "\n";
		}
		return os;
	}
#endif
};


//【線形方程式】O(n m min(n, m))
template <class T>
vector<T> gauss_jordan_elimination(const Matrix<T>& A, const vector<T>& b, vector<vector<T>>* xs = nullptr) {
	int n = A.n, m = A.m;

	// v : 拡大係数行列 (A | b)
	vector<vector<T>> v(n, vector<T>(m + 1));
	rep(i, n) rep(j, m) v[i][j] = A[i][j];
	rep(i, n) v[i][m] = b[i];

	// pivots[i] : 第 i 行のピボットが第何列にあるか
	vi pivots;

	// 注目位置を v[i][j] とする．
	int i = 0, j = 0;

	while (i < n && j <= m) {
		// 注目列の下方の行から非 0 成分を見つける．
		int i2 = i;
		while (i2 < n && v[i2][j] == T(0)) i2++;

		// 見つからなかったら注目位置を右に移す．
		if (i2 == n) { j++; continue; }

		// 見つかったら第 i 行とその行を入れ替える．
		if (i != i2) swap(v[i], v[i2]);

		// v[i][j] をピボットに選択する．
		pivots.push_back(j);

		// v[i][j] が 1 になるよう第 i 行全体を v[i][j] で割る．
		T vij_inv = T(1) / v[i][j];
		repi(j2, j, m) v[i][j2] *= vij_inv;

		// 第 i 行以外の第 j 列の成分が全て 0 になるよう第 i 行を定数倍して減じる．
		rep(i2, n) {
			if (v[i2][j] == T(0) || i2 == i) continue;

			T mul = v[i2][j];
			repi(j2, j, m) v[i2][j2] -= v[i][j2] * mul;
		}

		// 注目位置を右下に移す．
		i++; j++;
	}

	// 最後に見つかったピボットの位置が第 m 列ならば解なし．
	if (!pivots.empty() && pivots.back() == m) return vector<T>();

	// A x = b の特殊解 x0 の構成（任意定数は全て 0 にする）
	vector<T> x0(m);
	int rnk = sz(pivots);
	rep(i, rnk) x0[pivots[i]] = v[i][m];

	// 同次形 A x = 0 の一般解 {x} の基底の構成（任意定数を 1-hot にする）
	if (xs != nullptr) {
		xs->clear();

		int i = 0;
		rep(j, m) {
			if (i < rnk && j == pivots[i]) {
				i++;
				continue;
			}

			vector<T> x(m);
			x[j] = T(1);
			rep(i2, i) x[pivots[i2]] = -v[i2][j];
			xs->emplace_back(move(x));
		}
	}

	return x0;
}


// https://qiita.com/satoshin_astonish/items/a628ec64f29e77501d07
namespace satoshin {
	/* 内積 */
	double dot(const vl& x, const vd& y) {
		double z = 0.0;
		const int n = sz(x);
		for (int i = 0; i < n; ++i) z += x[i] * y[i];
		return z;
	}
	double dot(const vd& x, const vd& y) {
		double z = 0.0;
		const int n = sz(x);
		for (int i = 0; i < n; ++i) z += x[i] * y[i];
		return z;
	}
	double dot(const vl& x, const vl& y) {
		double z = 0.0;
		const int n = sz(x);
		for (int i = 0; i < n; ++i) z += x[i] * y[i];
		return z;
	}

	/* Gram-Schmidtの直交化 */
	tuple<vd, vvd> Gram_Schmidt_squared(const vvl& b) {
		const int n = sz(b), m = sz(b[0]); int i, j, k;
		vd B(n);
		vvd GSOb(n, vd(m)), mu(n, vd(n));
		for (i = 0; i < n; ++i) {
			mu[i][i] = 1.0;
			for (j = 0; j < m; ++j) GSOb[i][j] = (double)b[i][j];
			for (j = 0; j < i; ++j) {
				mu[i][j] = dot(b[i], GSOb[j]) / dot(GSOb[j], GSOb[j]);
				for (k = 0; k < m; ++k) GSOb[i][k] -= mu[i][j] * GSOb[j][k];
			}
			B[i] = dot(GSOb[i], GSOb[i]);
		}
		return std::forward_as_tuple(B, mu);
	}

	/* 部分サイズ基底簡約 */
	void SizeReduce(vvl& b, vvd& mu, const int i, const int j) {
		ll q;
		const int m = sz(b[0]);
		if (mu[i][j] > 0.5 || mu[i][j] < -0.5) {
			q = (ll)round(mu[i][j]);
			for (int k = 0; k < m; ++k) b[i][k] -= q * b[j][k];
			for (int k = 0; k <= j; ++k) mu[i][k] -= mu[j][k] * q;
		}
	}

	/* LLL基底簡約 */
	void LLLReduce(vvl& b, const float d = 0.99) {
		const int n = sz(b), m = sz(b[0]); int j, i, h;
		double t, nu, BB, C;
		auto [B, mu] = Gram_Schmidt_squared(b);
		ll tmp;
		for (int k = 1; k < n;) {
			h = k - 1;

			for (j = h; j > -1; --j) SizeReduce(b, mu, k, j);

			//Checks if the lattice basis matrix b satisfies Lovasz condition.
			if (k > 0 && B[k] < (d - mu[k][h] * mu[k][h]) * B[h]) {
				for (i = 0; i < m; ++i) { tmp = b[h][i]; b[h][i] = b[k][i]; b[k][i] = tmp; }

				nu = mu[k][h]; BB = B[k] + nu * nu * B[h]; C = 1.0 / BB;
				mu[k][h] = nu * B[h] * C; B[k] *= B[h] * C; B[h] = BB;

				for (i = 0; i <= k - 2; ++i) {
					t = mu[h][i]; mu[h][i] = mu[k][i]; mu[k][i] = t;
				}
				for (i = k + 1; i < n; ++i) {
					t = mu[i][k]; mu[i][k] = mu[i][h] - nu * t;
					mu[i][h] = t + mu[k][h] * mu[i][k];
				}

				--k;
			}
			else ++k;
		}
	}
}


vl LLLReduce(const vvm& lat_) {
	int h = sz(lat_);
	int w = sz(lat_[0]);

	vvl lat(h + w, vl(w));
	rep(i, h) rep(j, w) lat[i][j] = lat_[i][j].val();
	rep(i, w) lat[h + i][i] = mint::mod();
	h = sz(lat);
	satoshin::LLLReduce(lat);

	// L1 ノルムをチェックする．
	ll sum = 0;
	rep(j, w) sum += abs(lat[0][j]);
	dump("L1:", sum);

	// L1 ノルムが大きいものは捨てる．
	repi(i, 1, h - 1) {
		ll sum2 = 0;
		rep(j, w) sum2 += abs(lat[i][j]);

		if (sum2 > sum * 10.) {
			lat.resize(i);
			h = i;
			break;
		}
	}
	dump("lat:"); frac_print = 1; dumpel(lat); frac_print = 0;

	return lat[0];
}


// 変数係数線形漸化式の係数を計算し，埋め込み用のコードを出力する．
vvm embed_coefs_1D(const vm& seq, int TRM_ini = 1, int DEG_ini = 1, bool LLL = false) {
	int n = sz(seq);

	// TRM 項間の，係数多項式の次数 DEG 未満の変数係数線形漸化式
	//		Σt∈[0..TRM) Σd∈[0..DEG) coefs[t][d] (i-TRM+1+t)^d seq[i-t] = 0
	// を探す．
	int TRM = TRM_ini, DEG = DEG_ini;

	while (1) {
		//dump("TRM:", TRM, "DEG:", DEG);
		
		int h = n - TRM + 1;
		int w = TRM * DEG;

		// 行列方程式 A x = 0 を解いて一般解の基底 xs を求める．
		Matrix<mint> A(h, w);
		repi(i, TRM - 1, n - 1) {
			rep(t, TRM) rep(d, DEG) {
				A[i - TRM + 1][t * DEG + d] = mint(i - TRM + 1 + t).pow(d) * seq[i - t];
			}
		}
		
		vvm xs;
		gauss_jordan_elimination(A, vm(h), &xs);

		// 自明解 x = 0 しか存在しない場合は失敗．
		if (xs.empty()) {
			if (DEG == 1) {
				DEG = TRM + DEG;
				TRM = 1;
			}
			else {
				TRM++;
				DEG--;
			}
			continue;
		}

		dump("TRM:", TRM, "DEG:", DEG);
		dump("#eq:", h, "#var:", w);
		dump("xs:"); frac_print = 1; dumpel(xs); frac_print = 0;

		// 変数係数線形漸化式の係数
		vvm coefs(TRM, vm(DEG));

		if (LLL) {
			// A x = 0 の解空間の基底に LLL を適用する．
			// 性能はいまいちなのでガチでやるなら Mathematica を使う．
			auto lat0 = LLLReduce(xs);

			rep(t, TRM) rep(d, DEG) coefs[t][d] = lat0[t * DEG + d];
		}
		else {
			rep(t, TRM) rep(d, DEG) coefs[t][d] = xs.back()[t * DEG + d];
		}

		return coefs;
	}

	return vvm();
}


// 変数係数線形漸化式の係数を計算し，埋め込み用のコードを出力する．
pair<vvvm, vvvm> embed_coefs_2D(const vvm& tbl, int DEG1_ini = 1, int TRM2_ini = 1, int DEG2_ini = 1, bool LLL = false) {
	int n1 = sz(tbl);

	// TRM 項間の，係数多項式の次数 DEG 未満の変数係数線形漸化式
	//		Σt1∈[0..TRM1) Σd1∈[0..DEG1) Σt2∈[0..TRM2) Σd2∈[0..DEG2)
	//			c[t1][d1][t2][d2] (i1-TRM1+1+t1)^d1 (i2-TRM2+1+t2)^d2 tbl[i1-t1][i2-t2] = 0
	// を探す．
	int TRM1 = 1, DEG1 = DEG1_ini;
	int TRM2 = TRM2_ini, DEG2 = DEG2_ini;
	int P_MAX = max({ TRM1, DEG1, TRM2, DEG2 });

	while (1) {
		//dump("TRM1:", TRM1, "DEG1:", DEG1, "TRM2:", TRM2, "DEG2:", DEG2);
		
		int w = TRM1 * DEG1 * TRM2 * DEG2;

		// 行列方程式 A x = 0 を解いて一般解の基底 xs を求める．
		Matrix<mint> A(0, w);
		repi(i1, TRM1 - 1, n1 - 1) {
			int n2 = sz(tbl[i1]);
			repi(i2, TRM2 - 1, n2 - 1) {
				vm a(w); bool valid = true;
				rep(t1, TRM1) rep(d1, DEG1) rep(t2, TRM2) rep(d2, DEG2) {
					if (i2 - t2 >= sz(tbl[i1 - t1])) {
						valid = false;
						break;
					}
					int idx = ((t1 * DEG1 + d1) * TRM2 + t2) * DEG2 + d2;
					mint pow_i = mint(i1 - TRM1 + 1 + t1).pow(d1) * mint(i2 - TRM2 + 1 + t2).pow(d2);
					a[idx] = pow_i * tbl[i1 - t1][i2 - t2];
				}
				if (valid) A.push_back(a);
			}
		}
		int h = A.n;
		
		vvm xs;
		gauss_jordan_elimination(A, vm(h), &xs);

		// 自明解 x = 0 しか存在しない場合は失敗．
		if (xs.empty()) {
			while (1) {
				DEG2++;
				if (DEG2 > P_MAX) { DEG2 = 1; TRM2++; };
				if (TRM2 > P_MAX) { TRM2 = 1; DEG1++; };
				if (DEG1 > P_MAX) { DEG1 = 1; TRM1++; };
				if (TRM1 > 1) { TRM1 = 1; P_MAX++; };
				if (max({ TRM1, DEG1, TRM2, DEG2 }) == P_MAX) break;
			}
			continue;
		}

		dump("TRM1:", TRM1, "DEG1:", DEG1, "TRM2:", TRM2, "DEG2:", DEG2);
		dump("#eq:", h, "#var:", w);
		dump("xs:"); frac_print = 1; dumpel(xs); frac_print = 0;

		// 変数係数線形漸化式の係数
		vvvm coefs(DEG1, vvm(TRM2, vm(DEG2)));

		if (LLL) {
			// A x = 0 の解空間の基底に LLL を適用する．
			// 性能はいまいちなのでガチでやるなら Mathematica を使う．
			auto lat0 = LLLReduce(xs);

			rep(d1, DEG1) rep(t2, TRM2) rep(d2, DEG2) {
				int idx = (d1 * TRM2 + t2) * DEG2 + d2;
				coefs[d1][t2][d2] = lat0[idx];
			}
		}
		else {
			rep(d1, DEG1) rep(t2, TRM2) rep(d2, DEG2) {
				int idx = (d1 * TRM2 + t2) * DEG2 + d2;
				coefs[d1][t2][d2] = xs.back()[idx];
			}
		}

		// i1 方向への初項の延長
		dump("------- embed_coefs_1D -------");
		vvvm coefs1(TRM2 - 1);
		rep(i2, TRM2 - 1) {
			dump("--- i2:", i2, "---");

			vm seq; int offset = 0;
			rep(i1, sz(tbl)) {
				if (sz(tbl[i1]) <= i2) {
					if (offset == i1) {
						offset = i1 + 1;
						continue;
					} 
					else {
						break;
					}
				}
				seq.emplace_back(tbl[i1][i2]);
			}
			coefs1[i2] = embed_coefs_1D(seq, 1, 1, LLL);
		}

#ifdef _MSC_VER
		// 埋め込み用の文字列を出力する．
		auto to_signed_string = [](mint x) {
			int v = x.val();
			int mod = mint::mod();
			if (v > mod / 2) v -= mod;
			return to_string(v);
		};
		string eb;
		eb += "\n";
		eb += "constexpr int TRM1 = ";
		eb += to_string(TRM1);
		eb += ";\n";
		eb += "constexpr int DEG1 = ";
		eb += to_string(DEG1);
		eb += ";\n";
		eb += "constexpr int TRM2 = ";
		eb += to_string(TRM2);
		eb += ";\n";
		eb += "constexpr int DEG2 = ";
		eb += to_string(DEG2);
		eb += ";\n\n";

		eb += "vvm coefs1[TRM2 - 1] = {\n";
		rep(i2, TRM2 - 1) {
			eb += "{";
			rep(t, sz(coefs1[i2])) {
				eb += "{";
				rep(d, sz(coefs1[i2][t])) {
					eb += to_signed_string(coefs1[i2][t][d]) + ",";
				}
				eb.pop_back();
				eb += "},";
			}
			eb.pop_back();
			eb += "},\n";
		}
		eb.pop_back(); eb.pop_back();
		eb += "};\n\n";

		eb += "mint coefs[DEG1][TRM2][DEG2] = {\n";
		rep(d1, DEG1) {
			eb += "{";
			rep(t2, TRM2) {
				eb += "{";
				rep(d2, DEG2) {
					eb += to_signed_string(coefs[d1][t2][d2]) + ",";
				}
				eb.pop_back();
				eb += "},";
			}
			eb.pop_back();
			eb += "},\n";
		}
		eb.pop_back(); eb.pop_back();
		eb += "};\n";
		cout << eb;
#endif

		return { coefs1, coefs };
	}

	return pair<vvvm, vvvm>();
}


// 数列 seq を延長して seq[0..N] にする．
void solve_1D(vm& seq, int N, vvm coefs) {
	int TRM = sz(coefs);
	int DEG = sz(coefs[0]);

	int n = sz(seq);
	seq.resize(N + 1);

	// TRM 項間の，係数多項式の次数 DEG 未満の変数係数線形漸化式
	//		Σt∈[0..TRM) Σd∈[0..DEG) coefs[t][f] (i-TRM+1+t)^d a[i-t] = 0
	// を用いて数列 a を延長する．
	repi(i, n, N) {
		mint dnm = 0;
		mint pow_i = 1;
		rep(d, DEG) {
			dnm += coefs[0][d] * pow_i;
			pow_i *= i - TRM + 1;
		}

		mint num = 0;
		repi(t, 1, TRM - 1) {
			mint pow_i = 1;
			rep(d, DEG) {
				num += coefs[t][d] * pow_i * seq[i - t];
				pow_i *= i - TRM + 1 + t;
			}
		}

		// dnm * a[i] + num = 0 を解く．分母 0 に注意！
		if (dnm == 0) {
			dump("DIVISION BY ZERO at i =", i);
			Assert(dnm != 0);
		}
		seq[i] = -num / dnm;
	}
}


// 2 次元数列 tbl を元に seq = tbl[N][0..M] を計算する．
vm solve_2D(const vvm& tbl, int N, int M, vvvm coefs1, vvvm coefs) {
	int TRM1 = 1;
	int DEG1 = sz(coefs);
	int TRM2 = sz(coefs[0]);
	int DEG2 = sz(coefs[0][0]);
	
	vm res(TRM2 - 1);

	// i1 方向に tbl[..][0], ..., tbl[..][TRM2-2] を延長する．
	dump("------- solve_1D -------");
	rep(i2, TRM2 - 1) {
		dump("--- i2:", i2, "---");

		vm seq; int offset = 0;
		rep(i1, sz(tbl)) {
			if (sz(tbl[i1]) <= i2) {
				if (offset == i1) {
					offset = i1 + 1;
					continue;
				}
				else {
					break;
				}
			}
			seq.emplace_back(tbl[i1][i2]);
		}
		if (N - offset < 0) continue;

		solve_1D(seq, N - offset, coefs1[i2]);
		//dump("seq:", seq);

		res[i2] = seq[N - offset];
	}

	vm pow_i1s(DEG1);
	pow_i1s[0] = 1;
	repi(d1, 1, DEG1 - 1) pow_i1s[d1] = pow_i1s[d1 - 1] * (N - TRM1 + 1);

	// i2 方向に tbl[N][..] を延長する．
	res.resize(M + 1);
	repi(i2, TRM2 - 1, M) {
		mint dnm = 0;
		mint pow_i2 = 1;
		rep(d2, DEG2) {
			rep(d1, DEG1) {
				dnm += coefs[d1][0][d2] * pow_i1s[d1] * pow_i2;
			}
			pow_i2 *= i2 - TRM2 + 1;
		}

		mint num = 0;
		repi(t2, 1, TRM2 - 1) {
			mint pow_i2 = 1;
			rep(d2, DEG2) {
				rep(d1, DEG1) {
					num += coefs[d1][t2][d2] * pow_i1s[d1] * pow_i2 * res[i2 - t2];
				}
				pow_i2 *= i2 - TRM2 + 1 + t2;
			}
		}

		// dnm * tbl[N][i2] + num = 0 を解く．分母 0 に注意！
		if (dnm == 0) {
			dump("DIVISION BY ZERO at i1 =", N, "i2 =", i2);
			Assert(dnm != 0);
		}
		res[i2] = -num / dnm;
	}

	return res;
}



// 2 次元数列 tbl を元に seq = tbl[N][0..M] を計算する．
vm solve_2D(const vvm& tbl, int N, int M) {
	// --------------- embed_coefs() からの出力を貼る ----------------
	constexpr int TRM1 = 1;
	constexpr int DEG1 = 3;
	constexpr int TRM2 = 3;
	constexpr int DEG2 = 3;

	vvm coefs1[TRM2 - 1] = {
	{{0,499122176,499122176},{-1,0,1}},
	{{0,-499122176,499122176},{0,-2,1}} };

	mint coefs[DEG1][TRM2][DEG2] = {
	{{-10,-11,-3},{0,10,10},{6,1,-7}},
	{{4,-499122175,499122176},{-10,499122167,-499122175},{6,8,-1}},
	{{-499122176,-499122176,0},{499122176,499122175,0},{0,1,0}} };
	// --------------------------------------------------------------

	vm res(TRM2 - 1);

	// i1 方向に tbl[..][0], ..., tbl[..][TRM2-2] を延長する．
	dump("------- solve_1D -------");
	rep(i2, TRM2 - 1) {
		dump("--- i2:", i2, "---");

		vm seq; int offset = 0;
		rep(i1, sz(tbl)) {
			if (sz(tbl[i1]) <= i2) {
				if (offset == i1) {
					offset = i1 + 1;
					continue;
				}
				else {
					break;
				}
			}
			seq.emplace_back(tbl[i1][i2]);
		}
		if (N - offset < 0) continue;
		
		solve_1D(seq, N - offset, coefs1[i2]);
		//dump("seq:", seq);

		res[i2] = seq[N - offset];
	}

	vm pow_i1s(DEG1);
	pow_i1s[0] = 1;
	repi(d1, 1, DEG1 - 1) pow_i1s[d1] = pow_i1s[d1 - 1] * (N - TRM1 + 1);

	// i2 方向に tbl[N][..] を延長する．
	res.resize(M + 1);
	repi(i2, TRM2 - 1, M) {
		mint dnm = 0;		
		mint pow_i2 = 1;
		rep(d2, DEG2) {
			rep(d1, DEG1) {
				dnm += coefs[d1][0][d2] * pow_i1s[d1] * pow_i2;
			}
			pow_i2 *= i2 - TRM2 + 1;
		}

		mint num = 0;
		repi(t2, 1, TRM2 - 1) {
			mint pow_i2 = 1;
			rep(d2, DEG2) {
				rep(d1, DEG1) {
					num += coefs[d1][t2][d2] * pow_i1s[d1] * pow_i2 * res[i2 - t2];
				}
				pow_i2 *= i2 - TRM2 + 1 + t2;
			}
		}		

		// dnm * tbl[N][i2] + num = 0 を解く．分母 0 に注意！
		if (dnm == 0) {
			dump("DIVISION BY ZERO at i1 =", N, "i2 =", i2);
			Assert(dnm != 0);
		}
		res[i2] = -num / dnm;
	}

	return res;
}


vvm tbl = {
{0,0,0,0,0,0,0,0,0,0},
{1,0,0,0,0,0,0,0,0,0},
{3,3,0,0,0,0,0,0,0,0},
{8,9,8,0,0,0,0,0,0,0},
{20,24,24,20,0,0,0,0,0,0},
{48,60,64,60,48,0,0,0,0,0},
{112,144,160,160,144,112,0,0,0,0},
{256,336,384,400,384,336,256,0,0,0},
{576,768,896,960,960,896,768,576,0,0},
{1280,1728,2048,2240,2304,2240,2048,1728,1280,0} };


int main() {
//	input_from_file("input.txt");
//	output_to_file("output.txt");

	//【方法】
	// 愚直を書いて集めたデータをもとに変数係数線形漸化式を復元する．

	//【使い方】
	// 1. vm tbl = naive() を実装する．
	// 2. coefs = embed_coefs(tbl, TRM1_ini, DEG1_ini, TRM2_ini, DEG2_ini, LLL); を実行する．
	// 3. 出力を solve() 内に貼る．
	// 4. solve(tbl, n, m, [coefs]) で勝手に tbl[n][0..m] を求めてくれる．
	
	// 愚直解を用意する．再計算がイヤなら埋め込む．
	auto tbl = naive();
	
	// 愚直解を渡して変数係数線形漸化式の係数を得る．再計算がイヤなら埋め込む．
	// 引数：tbl, DEG1_ini, TRM2_ini, DEG2_ini, LLL?
	auto [coefs1, coefs] = embed_coefs_2D(tbl, 1, 1, 1, 0);

	int n;
	cin >> n;
				
	// 2 次元数列 tbl を元に seq = tbl[n][0..m] を計算する．
	// 整理すると綺麗な式になるなら FullSimplify[] すると速くなる．
	auto seq = solve_2D(tbl, n, n-1, coefs1, coefs);
//	auto seq = solve_2D(tbl, n, m);
	//dump(seq);
	
	vm a(n);
	cin >> a;

	mint res = 0;
	rep(i, n) res += a[i] * seq[i];

	EXIT(res);
}
/*
vvm tbl = {
{0,0,0,0,0,0,0,0,0,0},
{1,0,0,0,0,0,0,0,0,0},
{3,3,0,0,0,0,0,0,0,0},
{8,9,8,0,0,0,0,0,0,0},
{20,24,24,20,0,0,0,0,0,0},
{48,60,64,60,48,0,0,0,0,0},
{112,144,160,160,144,112,0,0,0,0},
{256,336,384,400,384,336,256,0,0,0},
{576,768,896,960,960,896,768,576,0,0},
{1280,1728,2048,2240,2304,2240,2048,1728,1280,0}};

TRM1: 1 DEG1: 3 TRM2: 3 DEG2: 3
#eq: 80 #var: 27
xs:
 0: 4 6 2 0 -5 -5 -2 -1 3 -4 -3 0 5 7 0 -1 -4 0 1 0 0 -2 0 0 1 0 0
 1: -10 -11 -3 0 10 10 6 1 -7 4 3/2 -1/2 -10 -19/2 3/2 6 8 -1 1/2 1/2 0 -1/2 -3/2 0 0 1 0
------- embed_coefs_1D -------
--- i2: 0 ---
TRM: 2 DEG: 3
#eq: 9 #var: 6
xs:
 0: 0 -1/2 -1/2 -1 0 1
--- i2: 1 ---
TRM: 2 DEG: 3
#eq: 9 #var: 6
xs:
 0: 0 1/2 -1/2 0 -2 1

constexpr int TRM1 = 1;
constexpr int DEG1 = 3;
constexpr int TRM2 = 3;
constexpr int DEG2 = 3;

vvm coefs1[TRM2 - 1] = {
{{0,499122176,499122176},{-1,0,1}},
{{0,-499122176,499122176},{0,-2,1}}};

mint coefs[DEG1][TRM2][DEG2] = {
{{-10,-11,-3},{0,10,10},{6,1,-7}},
{{4,-499122175,499122176},{-10,499122167,-499122175},{6,8,-1}},
{{-499122176,-499122176,0},{499122176,499122175,0},{0,1,0}}};
*/