Submission #72439796

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Source Code Expand

// 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


//【階乗など（法が大きな素数）】
/*
* Factorial_mint(int N) : O(n)
*	N まで計算可能として初期化する．
*
* mint fact(int n) : O(1)
*	n! を返す．
*
* mint fact_inv(int n) : O(1)
*	1/n! を返す（n が負なら 0 を返す）
*
* mint inv(int n) : O(1)
*	1/n を返す．
*
* mint inv_neg(int n) : O(1)
*	1/n を返す（n < 0 も可）
*
* mint perm(int n, int r) : O(1)
*	順列の数 nPr を返す．
*
* mint perm_inv(int n, int r) : O(1)
*	順列の数の逆数 1/nPr を返す．
*
* mint bin(int n, int r) : O(1)
*	二項係数 nCr を返す．
*
* mint bin_inv(int n, int r) : O(1)
*	二項係数の逆数 1/nCr を返す．
*
* mint mul(vi rs) : O(|rs|)
*	多項係数 nC[rs] を返す．（n = Σrs）
*
* mint hom(int n, int r) : O(1)
*	重複組合せの数 nHr = n+r-1Cr を返す（0H0 = 1 とする）
*
* mint neg_bin(int n, int r) : O(1)
*	負の二項係数 nCr = (-1)^r -n+r-1Cr を返す（n ≦ 0, r ≧ 0）
*
* mint pochhammer(int x, int n) : O(1)
*	ポッホハマー記号 x^(n) を返す（n ≧ 0）
*
* mint pochhammer_inv(int x, int n) : O(1)
*	ポッホハマー記号の逆数 1/x^(n) を返す（n ≧ 0）
*/
class Factorial_mint {
	int n_max;

	// 階乗と階乗の逆数の値を保持するテーブル
	vm fac, fac_inv;

public:
	// n! までの階乗とその逆数を前計算しておく．O(n)
	Factorial_mint(int n) : n_max(n), fac(n + 1), fac_inv(n + 1) {
		// verify : https://atcoder.jp/contests/dwacon6th-prelims/tasks/dwacon6th_prelims_b

		fac[0] = 1;
		repi(i, 1, n) fac[i] = fac[i - 1] * i;

		fac_inv[n] = fac[n].inv();
		repir(i, n - 1, 0) fac_inv[i] = fac_inv[i + 1] * (i + 1);
	}
	Factorial_mint() : n_max(0) {} // ダミー

	// n! を返す．
	mint fact(int n) const {
		// verify : https://atcoder.jp/contests/dwacon6th-prelims/tasks/dwacon6th_prelims_b

		Assert(0 <= n && n <= n_max);
		return fac[n];
	}

	// 1/n! を返す（n が負なら 0 を返す）
	mint fact_inv(int n) const {
		// verify : https://atcoder.jp/contests/abc289/tasks/abc289_h

		Assert(n <= n_max);
		if (n < 0) return 0;
		return fac_inv[n];
	}

	// 1/n を返す．
	mint inv(int n) const {
		// verify : https://atcoder.jp/contests/exawizards2019/tasks/exawizards2019_d

		Assert(n > 0);
		Assert(n <= n_max);
		return fac[n - 1] * fac_inv[n];
	}

	// 1/n を返す（n < 0 も可）
	mint inv_neg(int n) const {
		Assert(n != 0);
		Assert(abs(n) <= n_max);
		if (n > 0) return fac[n - 1] * fac_inv[n];
		else return -fac[-n - 1] * fac_inv[-n];
	}

	// 順列の数 nPr を返す．
	mint perm(int n, int r) const {
		// verify : https://atcoder.jp/contests/abc172/tasks/abc172_e

		Assert(n <= n_max);

		if (r < 0 || n - r < 0) return 0;
		return fac[n] * fac_inv[n - r];
	}

	// 順列の数 nPr の逆数を返す．
	mint perm_inv(int n, int r) const {
		// verify : https://yukicoder.me/problems/no/3139

		Assert(n <= n_max);
		Assert(0 <= r); Assert(r <= n);

		return fac_inv[n] * fac[n - r];
	}

	// 二項係数 nCr を返す．
	mint bin(int n, int r) const {
		// verify : https://judge.yosupo.jp/problem/binomial_coefficient_prime_mod

		Assert(n <= n_max);
		if (r < 0 || n - r < 0) return 0;
		return fac[n] * fac_inv[r] * fac_inv[n - r];
	}

	// 二項係数の逆数 1/nCr を返す．
	mint bin_inv(int n, int r) const {
		// verify : https://www.codechef.com/problems/RANDCOLORING

		Assert(n <= n_max);
		Assert(r >= 0);
		Assert(n - r >= 0);
		return fac_inv[n] * fac[r] * fac[n - r];
	}

	// 多項係数 nC[rs] を返す．
	mint mul(const vi& rs) const {
		// verify : https://yukicoder.me/problems/no/2141

		if (rs.empty()) return 1;

		if (*min_element(all(rs)) < 0) return 0;
		int n = accumulate(all(rs), 0);
		Assert(n <= n_max);

		mint res = fac[n];
		repe(r, rs) res *= fac_inv[r];

		return res;
	}

	// 重複組合せの数 nHr = n+r-1Cr を返す（0H0 = 1 とする）
	mint hom(int n, int r) {
		// verify : https://mojacoder.app/users/riantkb/problems/toj_ex_2

		if (n == 0) return (int)(r == 0);
		if (r < 0 || n - 1 < 0) return 0;
		Assert(n + r - 1 <= n_max);
		return fac[n + r - 1] * fac_inv[r] * fac_inv[n - 1];
	}

	// 負の二項係数 nCr を返す（n ≦ 0, r ≧ 0）
	mint neg_bin(int n, int r) {
		// verify : https://atcoder.jp/contests/abc345/tasks/abc345_g

		if (n == 0) return (int)(r == 0);
		if (r < 0 || -n - 1 < 0) return 0;
		Assert(-n + r - 1 <= n_max);
		return (r & 1 ? -1 : 1) * fac[-n + r - 1] * fac_inv[r] * fac_inv[-n - 1];
	}

	// ポッホハマー記号 x^(n) を返す（n ≧ 0）
	mint pochhammer(int x, int n) {
		// verify : https://atcoder.jp/contests/agc070/tasks/agc070_c

		int x2 = x + n - 1;
		if (x <= 0 && 0 <= x2) return 0;

		if (x > 0) {
			Assert(x2 <= n_max);
			return fac[x2] * fac_inv[x - 1];
		}
		else {
			Assert(-x <= n_max);
			return (n & 1 ? -1 : 1) * fac[-x] * fac_inv[-x2 - 1];
		}
	}

	// ポッホハマー記号の逆数 1/x^(n) を返す（n ≧ 0）
	mint pochhammer_inv(int x, int n) {
		// verify : https://atcoder.jp/contests/agc070/tasks/agc070_c

		int x2 = x + n - 1;
		Assert(!(x <= 0 && 0 <= x2));

		if (x > 0) {
			Assert(x2 <= n_max);
			return fac_inv[x2] * fac[x - 1];
		}
		else {
			Assert(-x <= n_max);
			return (n & 1 ? -1 : 1) * fac_inv[-x] * fac[-x2 - 1];
		}
	}
};
Factorial_mint fm((int)1e5 + 10);


// (i1, i2) = (n1, n2) に対する愚直解を返す．
mint naive_12(int n1, int n2) {
	mint res;

	return res;
}


// i1 = n1 に対する愚直解を返す．
vm naive_1(int n1) {
	vm res;
	
	return res;
}


// (i1,i2)∈[0..n1)×[0..n2) に対する愚直解を返す．
vvm naive() {
	int h = 60, w = 60;

	vvm c(h + 1, vm(w + 1));

	// 非交差経路 → LGV
	repi(i, 1, h) repi(j, 1, w) {
		int L = i + j - 2;
		c[i][j] = fm.bin(L, i - 1) * fm.bin(L, i - 1);
		c[i][j] -= fm.bin(L, i - 2) * fm.bin(L, i);
	}

	vvm dp(h + 1, vm(w + 1));
	dp[1][1] = dp[2][1] = dp[1][2] = 1;

	// 除原理，ちゃんとやりゃ速くなるけど P-recursive で殴るのでどうでもいい．
	repi(i, 2, h) repi(j, 2, w) {
		dp[i][j] = c[i][j];

		repi(i2, 0, i - 1) repi(j2, 0, j - 1) {
			if (i2 == 0 && j2 == 0) continue;

			dp[i][j] -= dp[i - i2][j - j2] * fm.bin(i - 2, i2) * fm.bin(j - 2, j2);
		}
	}

	vm pow2(h + w + 1);
	pow2[0] = 1;
	rep(i, h + w) pow2[i + 1] = pow2[i] * 2;

	// 隙間の挿入
	vvm dp2(h + 1, vm(w + 1));

	repi(i, 1, h) repi(j, 1, w) {
		repi(i2, 0, i) repi(j2, 0, j) {
			mint coef = 1;
			if (i2 > 0) coef *= fm.bin(i - 2, i2) * pow2[i2];
			if (j2 > 0) coef *= fm.bin(j - 2, j2) * pow2[j2];
			dp2[i][j] += dp[i - i2][j - j2] * coef;
		}
	}

	// ブロックの結合
	vvvm dp3(h + 1, vvm(w + 1, vm(2)));
	dp3[0][0][0] = 1;

	repi(i, 0, h) repi(j, 0, w) {
		repi(ni, i, h) repi(nj, j, w) {
			dp3[ni][nj][1] += dp3[i][j][0];
		}

		repi(ni, i + 1, h) repi(nj, j + 1, w) {
			dp3[ni][nj][0] += dp3[i][j][1] * dp2[ni - i][nj - j];
		}
	}

	vvm tbl(h + 1, vm(w + 1));
	repi(i, 0, h) repi(j, 0, w) tbl[i][j] = dp3[i][j][1];

	rep(i, h + 1) tbl[i].resize(i + 1);

#ifdef _MSC_VER
	// 埋め込み用
	string eb;
	eb += "vvm tbl = {\n";
	rep(i1, sz(tbl)) {
		eb += "{";
		rep(i2, sz(tbl[i1])) eb += to_string(tbl[i1][i2].val()) + ",";
		if (eb.back() == ',') eb.pop_back();
		eb += "},\n";
	}
	eb.pop_back(); eb.pop_back();
	eb += "};\n\n";
	cout << eb;
#endif

	return tbl;
}


// 累乗の前計算
constexpr int MAT_T = 5010;
constexpr int MAT_D = 20;
mint pows[MAT_T + 1][MAT_D + 1]; // pows[t][d] = t^d
void init_pows() {
	repi(t, 0, MAT_T) {
		pows[t][0] = 1;
		rep(d, MAT_D) pows[t][d + 1] = pows[t][d] * t;
	}
}


#ifndef HIDDEN_IN_VS // 折りたたみ用

//【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];
}


vl LLLReduce2(const vvm& xs) {
	int h = sz(xs);
	int w = sz(xs[0]);

	vl lat0(w);

#ifdef _MSC_VER
	string cmd;
	cmd += "wolframscript -code \"MOD=";
	cmd += to_string(mint::mod());
	cmd += ";";
	cmd += "SortBy[LatticeReduce@Join[{";
	rep(i, h) {
		cmd += "{";
		rep(j, w) {
			cmd += to_string(xs[i][j].val());
			cmd += ",";
		}
		if (cmd.back() == ',') cmd.pop_back();
		cmd += "},";
	}
	if (cmd.back() == ',') cmd.pop_back();
	cmd += "},MOD IdentityMatrix[";
	cmd += to_string(w);
	cmd += "]],N@Norm@# &]\"";
	//dump("cmd:", cmd);

	FILE* fp = _popen(cmd.c_str(), "r");
	char buf[1 << 20];
	while (fgets(buf, sizeof(buf), fp)) printf("%s", buf);
	_pclose(fp);

	stringstream ss{ buf + 2 };
	rep(j, w) {
		string s;
		getline(ss, s, ' ');
		lat0[j] = stol(s);
	}
#endif

	return lat0;
}


string to_signed_string(mint x) {
	int v = x.val();
	int mod = mint::mod();
	if (v > mod / 2) v -= mod;
	return to_string(v);
}


vm inv_all(int n) {
	vm inv(n + 1);

	constexpr int MOD = mint::mod();
	inv[1] = 1;
	repi(i, 2, n) {
		inv[i] = MOD - mint(MOD / i) * inv[MOD % i];
	}

	return inv;
}

#endif // 折りたたみ用


// 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, int DEG_ini, int LLL) {
	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;
	int P_MAX = max(TRM, DEG);
	
	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] = pows[i - TRM + 1 + t][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()) {
			while (1) {
				DEG++;
				if (DEG > P_MAX) { DEG = 1; TRM++; };
				if (TRM > P_MAX) { TRM = 1; P_MAX++; };
				if (max(TRM, DEG) == P_MAX) break;
			}
			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 == 0) {
			rep(t, TRM) rep(d, DEG) coefs[t][d] = xs.back()[t * DEG + d];
		}
		else if (LLL == 1) {
			// A x = 0 の解空間の基底に LLL を適用する．
			auto lat0 = LLLReduce(xs);
			rep(t, TRM) rep(d, DEG) coefs[t][d] = lat0[t * DEG + d];
		}
		else if (LLL == 2) {
			// A x = 0 の解空間の基底に本気の LLL を適用する（埋め込み専用）
			auto lat0 = LLLReduce2(xs);
			rep(t, TRM) rep(d, DEG) coefs[t][d] = lat0[t * DEG + d];
		}

		// 分母チェック
#ifdef _MSC_VER
		cout << "dnm 1D:" << endl;
		string cmd;
		cmd += "wolframscript -code \"MOD=";
		cmd += to_string(mint::mod());
		cmd += ";";
		cmd += "toFrac[x_]:=Module[{},Do[num=Mod[x*dnm,MOD,-MOD/2];If[Abs[num]<=Sqrt@MOD,Return[num/dnm,Module]],{dnm,1,Sqrt@MOD}]];";
		cmd += "Factor[";
		rep(d, DEG) {
			cmd += to_string(coefs[0][d].val()); // Modulus 指定しているのでこれでいい
			cmd += "*(i-";
			cmd += to_string(TRM);
			cmd += "+1)^";
			cmd += to_string(d);
			cmd += "+";
		}
		cmd.pop_back();
		cmd += ",Modulus->MOD]/.x_Integer:>toFrac[x]\"";
		//dump("cmd:", cmd);

		FILE* fp = _popen(cmd.c_str(), "r");
		char buf[1 << 16];
		while (fgets(buf, sizeof(buf), fp)) printf("%s", buf);
		_pclose(fp);
#endif

		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．
tuple<vvvm, vvvm, vvvvm> embed_coefs_2D(const vvm& tbl, int TRM1_ini, int DEG1_ini, int TRM2_ini, int DEG2_ini, int LLL) {
	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 = TRM1_ini, 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 = pows[i1 - TRM1 + 1 + t1][d1] * pows[i2 - TRM2 + 1 + t2][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 > P_MAX) { 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
		vvvvm coefs(TRM1, vvvm(DEG1, vvm(TRM2, vm(DEG2))));

		if (LLL == 0) {
			rep(t1, TRM1) rep(d1, DEG1) rep(t2, TRM2) rep(d2, DEG2) {
				int idx = ((t1 * DEG1 + d1) * TRM2 + t2) * DEG2 + d2;
				coefs[t1][d1][t2][d2] = xs.back()[idx];
			}
		}
		else if (LLL == 1) {
			// A x = 0 の解空間の基底に LLL を適用する．
			auto lat0 = LLLReduce(xs);
			rep(t1, TRM1) rep(d1, DEG1) rep(t2, TRM2) rep(d2, DEG2) {
				int idx = ((t1 * DEG1 + d1) * TRM2 + t2) * DEG2 + d2;
				coefs[t1][d1][t2][d2] = lat0[idx];
			}
		}
		else if (LLL == 2) {
			// A x = 0 の解空間の基底に本気の LLL を適用する（埋め込み専用）
			auto lat0 = LLLReduce2(xs);
			rep(t1, TRM1) rep(d1, DEG1) rep(t2, TRM2) rep(d2, DEG2) {
				int idx = ((t1 * DEG1 + d1) * TRM2 + t2) * DEG2 + d2;
				coefs[t1][d1][t2][d2] = lat0[idx];
			}
		}

		// 分母チェック
#ifdef _MSC_VER
		cout << "dnm 2D:" << endl;
		string cmd;
		cmd += "wolframscript -code \"MOD=";
		cmd += to_string(mint::mod());
		cmd += ";";
		cmd += "toFrac[x_]:=Module[{},Do[num=Mod[x*dnm,MOD,-MOD/2];If[Abs[num]<=Sqrt@MOD,Return[num/dnm,Module]],{dnm,1,Sqrt@MOD}]];";
		cmd += "Factor[";
		rep(d2, DEG2) {
			rep(d1, DEG1) {
				cmd += to_signed_string(coefs[0][d1][0][d2]);
				cmd += "*(i1-";
				cmd += to_string(TRM1);
				cmd += "+1)^";
				cmd += to_string(d1);
				cmd += "*(i2-";
				cmd += to_string(TRM2);
				cmd += "+1)^";
				cmd += to_string(d2);
				cmd += "+";
			}
		}
		cmd.pop_back();
		cmd += "]/.x_Integer:>toFrac[x]\""; // 多変数では Modulus は指定できない
		//dump("cmd:", cmd);

		FILE* fp = _popen(cmd.c_str(), "r");
		char buf[1 << 16];
		while (fgets(buf, sizeof(buf), fp)) printf("%s", buf);
		_pclose(fp);
#endif

		// i1 方向への初項の延長
		dump("------- embed_coefs_1D_1 -------");
		vvvm coefs1(TRM2 - 1);
		rep(i2, TRM2 - 1) {
			dump("--- i2:", i2, "---");

			vm seq;
			repi(i1, i2, INF) {
				if (sz(tbl) <= i1 || sz(tbl[i1]) <= i2) break;
				seq.emplace_back(tbl[i1][i2]);
			}
			coefs1[i2] = embed_coefs_1D(seq, 1, 1, LLL);

			// デフォルトでは良くない漸化式が復元されてしまう場合はここで対処．
			//if (i2 == 2) coefs1[i2] = embed_coefs_1D(seq, 3, 1, LLL);
		}

		// i1+i2 方向への初項の延長
		dump("------- embed_coefs_1D_2 -------");
		vvvm coefs2(TRM1 - 1);
		rep(i1, TRM1 - 1) {
			dump("--- i1:", i1, "---");

			vm seq;
			rep(i2, INF) {
				if (sz(tbl) <= i1 + i2 || sz(tbl[i1 + i2]) <= i2) break;
				seq.emplace_back(tbl[i1 + i2][i2]);
			}
			coefs2[i1] = embed_coefs_1D(seq, 1, 1, LLL);

			// デフォルトでは良くない漸化式が復元されてしまう場合はここで対処．
			//if (i1 == 2) coefs2[i1] = embed_coefs_1D(seq, 3, 1, LLL);
		}

#ifdef _MSC_VER
		// 埋め込み用の文字列を出力する．
		string eb = "\n";

		eb += "vvm res(N+1);repi(i,0,N)res[i].resize(i+1);\n";

		rep(i2, TRM2 - 1) {
			vvm coefs = coefs1[i2];
			int TRM = sz(coefs);
			int DEG = sz(coefs[0]);

			vm seq;
			repi(i1, i2, INF) {
				if (sz(tbl) <= i1 || sz(tbl[i1]) <= i2) break;
				seq.emplace_back(tbl[i1][i2]);
			}

			eb += "if(" + to_string(i2) + "<=N){\n";

			eb += "vm seq{";
			repe(x, seq) {
				eb += to_signed_string(x) + ",";
			}
			eb.pop_back();
			eb += "};";
			eb += "seq.resize(N+1-" + to_string(i2) + ");\n";

			eb += "repi(i," + to_string(sz(seq)) + ",N-" + to_string(i2) + "){\n";
			eb += "mint num=0,dnm=0,tmp;\n";
			repir(t, TRM - 1, 0) {
				if (t == 0) eb += "\n";
				eb += "tmp=0;";								
				rep(d, DEG) {
					if (coefs[t][d] == 0) continue;
					eb += "tmp+=";
					eb += to_signed_string(coefs[t][d]);
					if (d == 1) eb += "*mint(i-" + to_string(TRM - 1 - t) + ")";
					else if (d >= 2) eb += "*pows[(i-" + to_string(TRM - 1 - t) + ")][" + to_string(d) + "]";
					eb += ";";
				}
				if (t == 0) eb += "dnm+=tmp;";
				else eb += "num+=tmp*seq[i-" + to_string(t) + "];";
			}
			eb += "\n";
			eb += "if(dnm==0){";
			eb += "dump(\"DIVISION BY ZERO at i1 =\",i,\"i2 = " + to_string(i2) + "\");";
			eb += "Assert(dnm!=0);";
			eb += "}\n";
			eb += "seq[i]=-num/dnm;\n";
			eb += "}\n";

			eb += "repi(i,0,N-" + to_string(i2) + ")res[i+" + to_string(i2) + "][" + to_string(i2) + "]=seq[i];\n";

			eb += "}\n";
		}

		rep(i1, TRM1 - 1) {
			vvm coefs = coefs2[i1];
			int TRM = sz(coefs);
			int DEG = sz(coefs[0]);

			vm seq;
			rep(i2, INF) {
				if (sz(tbl) <= i1 + i2 || sz(tbl[i1 + i2]) <= i2) break;
				seq.emplace_back(tbl[i1 + i2][i2]);
			}

			eb += "if(" + to_string(i1) + "<=N){\n";

			eb += "vm seq{";
			repe(x, seq) {
				eb += to_signed_string(x) + ",";
			}
			eb.pop_back();
			eb += "};";
			eb += "seq.resize(N+1-" + to_string(i1) + ");\n";

			eb += "repi(i," + to_string(sz(seq)) + ",N-" + to_string(i1) + "){\n";
			eb += "mint num=0,dnm=0,tmp;\n";
			repir(t, TRM - 1, 0) {
				if (t == 0) eb += "\n";
				eb += "tmp=0;";
				rep(d, DEG) {
					if (coefs[t][d] == 0) continue;
					eb += "tmp+=";
					eb += to_signed_string(coefs[t][d]);
					if (d == 1) eb += "*mint(i-" + to_string(TRM - 1 - t) + ")";
					else if (d >= 2) eb += "*pows[(i-" + to_string(TRM - 1 - t) + ")][" + to_string(d) + "]";
					eb += ";";
				}
				if (t == 0) eb += "dnm+=tmp;";
				else eb += "num+=tmp*seq[i-" + to_string(t) + "];";
			}
			eb += "\n";
			eb += "if(dnm==0){";
			eb += "dump(\"DIVISION BY ZERO at i1 =\",i+" + to_string(i1) + ",\"i2 =\",i);";
			eb += "Assert(dnm!=0);";
			eb += "}\n";
			eb += "seq[i]=-num/dnm;\n";
			eb += "}\n";

			eb += "repi(i,0,N-" + to_string(i1) + ")res[i+" + to_string(i1) + "][i]=seq[i];\n";

			eb += "}\n";
		}
		
		eb += "repi(i1," + to_string(TRM1 + TRM2 - 2) + ",N)";
		eb += "repi(i2," + to_string(TRM2 - 1) + ",i1-" + to_string(TRM1 - 1) + "){\n";
		eb += "mint num=0,dnm=0,tmp;\n";
		repir(t1, TRM1 - 1, 0) repir(t2, TRM2 - 1, 0) {
			if (t1 == 0 && t2 == 0) eb += "\n";
			eb += "tmp=0;";
			rep(d1, DEG1) rep(d2, DEG2) {
				if (coefs[t1][d1][t2][d2] == 0) continue;
				eb += "tmp+=";
				eb += to_signed_string(coefs[t1][d1][t2][d2]);
				if (d1 == 1) eb += "*mint(i1-" + to_string(TRM1 - 1 - t1) + ")";
				else if (d1 >= 2) eb += "*pows[i1-" + to_string(TRM1 - 1 - t1) + "][" + to_string(d1) + "]";
				if (d2 == 1) eb += "*mint(i2-" + to_string(TRM2 - 1 - t2) + ")";
				else if (d2 >= 2) eb += "*pows[i2-" + to_string(TRM2 - 1 - t2) + "][" + to_string(d2) + "]";
				eb += ";";
			}
			if (t1 == 0 && t2 == 0) eb += "dnm+=tmp;";
			else eb += "num+=tmp*res[i1-" + to_string(t1) + "][i2-" + to_string(t2) + "];";
		}
		eb += "\n";
		eb += "if(dnm==0){";
		eb += "dump(\"DIVISION BY ZERO at i1 =\",i1,\"i2 =\",i2);";
		eb += "Assert(dnm!=0);";
		eb += "}\n";
		eb += "res[i1][i2]=-num/dnm;\n";
		eb += "}\n";

		eb += "return res;\n\n";

		cout << eb;
#endif

		return { coefs1, coefs2, coefs };
	}

	return tuple<vvvm, vvvm, vvvvm>();
}


// 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;
		rep(d, DEG) {
			dnm += coefs[0][d] * pows[i - TRM + 1][d];
		}

		mint num = 0;
		repi(t, 1, TRM - 1) {
			rep(d, DEG) {
				num += coefs[t][d] * pows[i - TRM + 1 + t][d] * seq[i - 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[0..N][0..N] の下三角部分を計算する．
vvm solve_2D(const vvm& tbl, int N, const vvvm& coefs1, const vvvm& coefs2, const vvvvm& coefs) {
	int TRM1 = sz(coefs);
	int DEG1 = sz(coefs[0]);
	int TRM2 = sz(coefs[0][0]);
	int DEG2 = sz(coefs[0][0][0]);
	
	vvm res(N + 1);
	repi(i, 0, N) res[i].resize(i + 1);

	// i1 方向に初項を延長する．
	dump("------- solve_1D_1 -------");
	rep(i2, TRM2 - 1) {
		dump("--- i2:", i2, "---");
		if (N - i2 < 0) continue;

		vm seq;
		repi(i1, i2, INF) {
			if (sz(tbl) <= i1 || sz(tbl[i1]) <= i2) break;
			seq.emplace_back(tbl[i1][i2]);
		}
		
		solve_1D(seq, N - i2, coefs1[i2]);
		//dump("seq:", seq);

		repi(i1, 0, N - i2) {
			res[i1 + i2][i2] = seq[i1];
		}
	}

	// i1+i2 方向に初項を延長する．
	dump("------- solve_1D_2 -------");
	rep(i1, TRM1 - 1) {
		dump("--- i1:", i1, "---");
		if (N - i1 < 0) continue;

		vm seq;
		rep(i2, INF) {
			if (sz(tbl) <= i1 + i2 || sz(tbl[i1 + i2]) <= i2) break;
			seq.emplace_back(tbl[i1 + i2][i2]);
		}
		
		solve_1D(seq, N - i1, coefs2[i1]);
		//dump("seq:", seq);

		repi(i2, 0, N - i1) {
			res[i1 + i2][i2] = seq[i2];
		}
	}
	
	// 平面的に延長する．
	repi(i1, TRM1 + TRM2 - 2, N) repi(i2, TRM2 - 1, i1 - TRM1 + 1) {
		mint num = 0, dnm = 0;
		rep(t1, TRM1) rep(t2, TRM2) {
			if (t1 == 0 && t2 == 0) {
				rep(d2, DEG2) rep(d1, DEG1) {
					mint p = pows[i1 - TRM1 + 1 + t1][d1] * pows[i2 - TRM2 + 1 + t2][d2];
					dnm += coefs[t1][d1][t2][d2] * p;
				}
			}
			else {
				rep(d2, DEG2) rep(d1, DEG1) {
					mint p = pows[i1 - TRM1 + 1 + t1][d1] * pows[i2 - TRM2 + 1 + t2][d2];
					assert((i2 - t2) <= (i1 - t1));
					num += coefs[t1][d1][t2][d2] * p * res[i1 - t1][i2 - 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 =", i1, "i2 =", i2);
			Assert(dnm != 0);
		}
		res[i1][i2] = -num / dnm;
	}

	return res;
}


// 2 次元数列 tbl を元に seq = tbl[0..N][0..N] の下三角部分を計算する．
vvm solve_2D(int N) {
	// 除算回避用
	//auto inv = inv_all(N + 10);
	/*
	mint dnm_inv = inv[i];
	seq[i] = -num * dnm_inv;
	mint dnm_inv = inv[i2];
	res[i1][i2] = -num * dnm_inv;
	*/

	// --------------- embed_coefs() からの出力を貼る ----------------
	vvm res(N + 1); repi(i, 0, N)res[i].resize(i + 1);
	if (0 <= N) {
		vm seq{ 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 }; seq.resize(N + 1 - 0);
		repi(i, 61, N - 0) {
			mint num = 0, dnm = 0, tmp;
			tmp = 0; tmp += 1; num += tmp * seq[i - 1];
			tmp = 0; tmp += -1; dnm += tmp;
			if (dnm == 0) { dump("DIVISION BY ZERO at i1 =", i, "i2 = 0"); Assert(dnm != 0); }
			seq[i] = -num / dnm;
		}
		repi(i, 0, N - 0)res[i + 0][0] = seq[i];
	}
	if (1 <= N) {
		vm seq{ 2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,131072,262144,524288,1048576,2097152,4194304,8388608,16777216,33554432,67108864,134217728,268435456,-461373441,75497471,150994942,301989884,-394264585,209715183,419430366,-159383621,-318767242,360709869,-276824615,444595123,-109054107,-218108214,-436216428,125811497,251622994,-494998365,8247623,16495246,32990492,65980984,131961968,263923936,-470396481,57451391,114902782,229805564,459611128,-79022097,-158044194,-316088388 }; seq.resize(N + 1 - 1);
		repi(i, 60, N - 1) {
			mint num = 0, dnm = 0, tmp;
			tmp = 0; tmp += 1; num += tmp * seq[i - 1];
			tmp = 0; tmp += 499122176; dnm += tmp;
			if (dnm == 0) { dump("DIVISION BY ZERO at i1 =", i, "i2 = 1"); Assert(dnm != 0); }
			seq[i] = -num / dnm;
		}
		repi(i, 0, N - 1)res[i + 1][1] = seq[i];
	}
	if (0 <= N) {
		vm seq{ 1,2,13,147,2103,34088,597539,11055950,212833803,230491390,-1961723,383222689,-472600503,407771542,82041758,-445706790,360894853,135478333,238792991,-178773677,-254651904,-45007292,-68749064,-44433988,-401377151,114265380,-419476184,-58887970,-203997975,425343623,135253218,-99050051,494863004,-192028754,-154349791,-97122476,-75350262,-264112780,313646564,25796115,98973101,-323915999,-265981017,429417612,369179625,123266929,-222936942,14709711,393704450,-301623745,325153793,-68586895,248782876,-202077810,187601692,-396080085,-450407895,-497525714,212902040,127158484,-392059194 }; seq.resize(N + 1 - 0);
		repi(i, 61, N - 0) {
			mint num = 0, dnm = 0, tmp;
			tmp = 0; tmp += 150600; tmp += -110990 * mint(i - 0); tmp += 28199 * pows[(i - 0)][2]; tmp += -2530 * pows[(i - 0)][3]; tmp += 1 * pows[(i - 0)][6]; num += tmp * seq[i - 6]; tmp = 0; tmp += -188343444; tmp += 462103570 * mint(i - 1); tmp += 391330131 * pows[(i - 1)][2]; tmp += 252190034 * pows[(i - 1)][3]; tmp += 240626235 * pows[(i - 1)][4]; tmp += 224306371 * pows[(i - 1)][5]; tmp += -302180014 * pows[(i - 1)][6]; num += tmp * seq[i - 5]; tmp = 0; tmp += 396476159; tmp += -65136719 * mint(i - 2); tmp += 49582361 * pows[(i - 2)][2]; tmp += 374715223 * pows[(i - 2)][3]; tmp += 380397584 * pows[(i - 2)][4]; tmp += 116414486 * pows[(i - 2)][5]; tmp += -254197061 * pows[(i - 2)][6]; num += tmp * seq[i - 4]; tmp = 0; tmp += 253651428 * mint(i - 3); tmp += -177203373 * pows[(i - 3)][2]; tmp += -25785211 * pows[(i - 3)][3]; tmp += -299167339 * pows[(i - 3)][4]; tmp += 42787621 * pows[(i - 3)][5]; tmp += 11507200 * pows[(i - 3)][6]; num += tmp * seq[i - 3]; tmp = 0; tmp += -3529310; tmp += -492974598 * mint(i - 4); tmp += -168201235 * pows[(i - 4)][2]; tmp += 251724327 * pows[(i - 4)][3]; tmp += -143228435 * pows[(i - 4)][4]; tmp += -432570044 * pows[(i - 4)][5]; tmp += -247435514 * pows[(i - 4)][6]; num += tmp * seq[i - 2]; tmp = 0; tmp += -151286888; tmp += -136918737 * mint(i - 5); tmp += -312150732 * pows[(i - 5)][2]; tmp += -333879573 * pows[(i - 5)][3]; tmp += -447357106 * pows[(i - 5)][4]; tmp += 31021132 * pows[(i - 5)][5]; tmp += -133128514 * pows[(i - 5)][6]; num += tmp * seq[i - 1];
			tmp = 0; tmp += -268836636; tmp += 342717524 * mint(i - 6); tmp += 51661427 * pows[(i - 6)][2]; tmp += 171694261 * pows[(i - 6)][3]; tmp += 378408904 * pows[(i - 6)][4]; tmp += -95171582 * pows[(i - 6)][5]; tmp += -72810451 * pows[(i - 6)][6]; dnm += tmp;
			if (dnm == 0) { dump("DIVISION BY ZERO at i1 =", i + 0, "i2 =", i); Assert(dnm != 0); }
			seq[i] = -num / dnm;
		}
		repi(i, 0, N - 0)res[i + 0][i] = seq[i];
	}
	if (1 <= N) {
		vm seq{ 1,4,38,506,7887,134825,2451038,46570858,-83389524,477938240,479393490,205394185,296687504,457784290,-254115804,73837149,78289056,461375479,-358276184,462838948,-343301175,99505652,-366090003,-323510595,-225426833,-187506638,261441219,-174649998,-182162252,56767261,452030203,-146211959,-133506527,-2895216,-146263276,484079919,-474532025,-353424966,290080234,-200903754,-72017877,452004736,428314365,-118994324,-374831619,-420831057,-116438429,368640185,-170996704,117964471,-486833110,176311955,148131958,-213818864,-227131262,-117616729,-419435633,-464906123,-125611790,121443279 }; seq.resize(N + 1 - 1);
		repi(i, 60, N - 1) {
			mint num = 0, dnm = 0, tmp;
			tmp = 0; tmp += 71520526; tmp += 382502251 * mint(i - 0); tmp += 493789538 * pows[(i - 0)][2]; tmp += -261960480 * pows[(i - 0)][3]; tmp += 348439154 * pows[(i - 0)][4]; tmp += -366622030 * pows[(i - 0)][5]; tmp += 1 * pows[(i - 0)][6]; num += tmp * seq[i - 6]; tmp = 0; tmp += 118719338; tmp += 332362566 * mint(i - 1); tmp += -233377927 * pows[(i - 1)][2]; tmp += 82565809 * pows[(i - 1)][3]; tmp += 305822346 * pows[(i - 1)][4]; tmp += -444291400 * pows[(i - 1)][5]; tmp += -13569222 * pows[(i - 1)][6]; num += tmp * seq[i - 5]; tmp = 0; tmp += 464599253; tmp += 269961708 * mint(i - 2); tmp += 45174028 * pows[(i - 2)][2]; tmp += -401628523 * pows[(i - 2)][3]; tmp += -135312413 * pows[(i - 2)][4]; tmp += -335145520 * pows[(i - 2)][5]; tmp += 61512137 * pows[(i - 2)][6]; num += tmp * seq[i - 4]; tmp = 0; tmp += 215875310; tmp += -44959087 * mint(i - 3); tmp += 78852102 * pows[(i - 3)][2]; tmp += 434928673 * pows[(i - 3)][3]; tmp += -449685771 * pows[(i - 3)][4]; tmp += -361408845 * pows[(i - 3)][5]; tmp += 86714126 * pows[(i - 3)][6]; num += tmp * seq[i - 3]; tmp = 0; tmp += 266704007; tmp += -437475369 * mint(i - 4); tmp += 269868829 * pows[(i - 4)][2]; tmp += -473695691 * pows[(i - 4)][3]; tmp += -121358614 * pows[(i - 4)][4]; tmp += -302752049 * pows[(i - 4)][5]; tmp += 341549435 * pows[(i - 4)][6]; num += tmp * seq[i - 2]; tmp = 0; tmp += -146677883; tmp += -326067800 * mint(i - 5); tmp += 490292816 * pows[(i - 5)][2]; tmp += -320402306 * pows[(i - 5)][3]; tmp += 359616910 * pows[(i - 5)][4]; tmp += -209997345 * pows[(i - 5)][5]; tmp += -137194717 * pows[(i - 5)][6]; num += tmp * seq[i - 1];
			tmp = 0; tmp += -151537631; tmp += 148961201 * mint(i - 6); tmp += -155062007 * pows[(i - 6)][2]; tmp += -171530394 * pows[(i - 6)][3]; tmp += -103793277 * pows[(i - 6)][4]; tmp += 121696129 * pows[(i - 6)][5]; tmp += -339011760 * pows[(i - 6)][6]; dnm += tmp;
			if (dnm == 0) { dump("DIVISION BY ZERO at i1 =", i + 1, "i2 =", i); Assert(dnm != 0); }
			seq[i] = -num / dnm;
		}
		repi(i, 0, N - 1)res[i + 1][i] = seq[i];
	}
	repi(i1, 4, N)repi(i2, 2, i1 - 2) {
		mint num = 0, dnm = 0, tmp;
		tmp = 0; num += tmp * res[i1 - 2][i2 - 2]; tmp = 0; tmp += 266198496; tmp += -199648871 * mint(i2 - 1); tmp += -266198496 * mint(i1 - 0); tmp += 199648870 * mint(i1 - 0) * mint(i2 - 1); tmp += 1 * pows[i1 - 0][2] * mint(i2 - 1); num += tmp * res[i1 - 2][i2 - 1]; tmp = 0; tmp += -66549623; tmp += 66549623 * mint(i1 - 0); tmp += -399297740 * mint(i1 - 0) * mint(i2 - 2); tmp += 399297740 * pows[i1 - 0][2] * mint(i2 - 2); num += tmp * res[i1 - 2][i2 - 0]; tmp = 0; tmp += -133099248; tmp += 332748118 * mint(i2 - 0); tmp += -199648870 * pows[i2 - 0][2]; tmp += 2 * mint(i1 - 1); tmp += -3 * mint(i1 - 1) * mint(i2 - 0); tmp += 1 * mint(i1 - 1) * pows[i2 - 0][2]; num += tmp * res[i1 - 1][i2 - 2]; tmp = 0; tmp += 399297742; tmp += -199648874 * mint(i2 - 1); tmp += -199648872 * pows[i2 - 1][2]; tmp += 199648875 * mint(i1 - 1); tmp += -2 * mint(i1 - 1) * mint(i2 - 1); tmp += -2 * mint(i1 - 1) * pows[i2 - 1][2]; tmp += 199648871 * pows[i1 - 1][2]; tmp += -2 * pows[i1 - 1][2] * mint(i2 - 1); num += tmp * res[i1 - 1][i2 - 1]; tmp = 0; tmp += 266198496; tmp += 66549627 * mint(i2 - 2); tmp += 399297742 * pows[i2 - 2][2]; tmp += 399297738 * mint(i1 - 1); tmp += 199648874 * mint(i1 - 1) * mint(i2 - 2); tmp += -399297740 * mint(i1 - 1) * pows[i2 - 2][2]; tmp += -399297742 * pows[i1 - 1][2]; tmp += 199648872 * pows[i1 - 1][2] * mint(i2 - 2); num += tmp * res[i1 - 1][i2 - 0]; tmp = 0; tmp += -465847369; tmp += 66549631 * mint(i2 - 0); tmp += 399297738 * pows[i2 - 0][2]; tmp += -199648873 * mint(i1 - 2); tmp += -199648867 * mint(i1 - 2) * mint(i2 - 0); tmp += 399297740 * mint(i1 - 2) * pows[i2 - 0][2]; num += tmp * res[i1 - 0][i2 - 2]; tmp = 0; tmp += 465847361; tmp += 399297750 * mint(i2 - 1); tmp += 4 * pows[i2 - 1][2]; tmp += -332748121 * mint(i1 - 2); tmp += 199648876 * mint(i1 - 2) * mint(i2 - 1); tmp += 199648872 * mint(i1 - 2) * pows[i2 - 1][2]; tmp += -199648871 * pows[i1 - 2][2]; tmp += -399297740 * pows[i1 - 2][2] * mint(i2 - 1); num += tmp * res[i1 - 0][i2 - 1];
		tmp = 0; tmp += -332748119; tmp += -465847371 * mint(i2 - 2); tmp += 399297740 * pows[i2 - 2][2]; tmp += -465847363 * mint(i1 - 2); tmp += -399297744 * mint(i1 - 2) * mint(i2 - 2); tmp += -199648871 * mint(i1 - 2) * pows[i2 - 2][2]; tmp += 199648871 * pows[i1 - 2][2]; tmp += -199648871 * pows[i1 - 2][2] * mint(i2 - 2); dnm += tmp;
		if (dnm == 0) { dump("DIVISION BY ZERO at i1 =", i1, "i2 =", i2); Assert(dnm != 0); }
		res[i1][i2] = -num / dnm;
	}
	return res;
	// --------------------------------------------------------------	
}


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. vvm tbl = naive() を実装する．
	// 2. embed_coefs2D() を実行する．
	// 3. 出力を solve() 内に貼る．
	// 4. solve(tbl, n) で勝手に tbl[0..n][0..n] の下三角部分を求めてくれる．
	
	// 累乗テーブルの初期化
	init_pows();

	// 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, TRM1_ini, DEG1_ini, TRM2_ini, DEG2_ini, LLL
//	auto [coefs1, coefs2, coefs] = embed_coefs_2D(tbl, 1, 1, 1, 1, 0);
	
	int n, m;
	cin >> n >> m;

	if (n < m) swap(n, m);
		
	// 下三角行列 tbl を延長して tbl[0..n][0..n] の下三角部分を求めてくれる．
	// 整理すると綺麗な式になるなら FullSimplify[] すると速くなる．
//	auto dp = solve_2D(tbl, n, coefs1, coefs2, coefs);
	auto dp = solve_2D(n);
//	dumpel(dp);
	
	mint res = dp[n][m];
	
	EXIT(res);
}

Submission Info

Compile Error

./Main.cpp: In function 'mint naive_12(int, int)':
./Main.cpp:322:19: warning: unused parameter 'n1' [-Wunused-parameter]
  322 | mint naive_12(int n1, int n2) {
      |               ~~~~^~
./Main.cpp:322:27: warning: unused parameter 'n2' [-Wunused-parameter]
  322 | mint naive_12(int n1, int n2) {
      |                       ~~~~^~
./Main.cpp: In function 'vm naive_1(int)':
./Main.cpp:330:16: warning: unused parameter 'n1' [-Wunused-parameter]
  330 | vm naive_1(int n1) {
      |            ~~~~^~
./Main.cpp: In function 'vl LLLReduce2(const vvm&)':
./Main.cpp:755:13: warning: unused variable 'h' [-Wunused-variable]
  755 |         int h = sz(xs);
      |             ^

Judge Result