#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, i3) = (n1, n2, n3) に対する愚直解を返す．
mint naive_123(int i0, int i1, int i2) {
	int a1 = i0 + 1;
	int a2 = a1 + i1 + 1;
	int n = a2 + i2 + 1;

	if (a1 == a2) return fm.fact(n) * n;

	if (a1 > a2) swap(a1, a2);

	// とりあえず多項式オーダーにした
	
	int cL = a1;
	int cM = a2 - a1;
	int cR = n - a2;
	//dump(cL, cM, cR);

	mint res = 0;

	//dump("a2 より大，不動 :");
	repi(i, 1, n) {
		repi(cLl, 0, cL) repi(cMl, 0, cM) {
			int cLr = cL - cLl;
			int cMr = cM - cMl;
			int cRl = (i - 1) - cLl - cMl;
			int cRr = (n - i) - cLr - cMr;
			if (cRl < 0 || cRr < 0) continue;
			if (cMl + cRl < cLr) continue;
			if (cRl < cLr + cMr) continue;

			mint pos = fm.mul({ cLl, cMl, cRl }) * fm.mul({ cLr, cMr, cRr });
			mint val = fm.fact(cL) * fm.fact(cM) * fm.fact(cR - 1);
			//dump("i:", i, "c:", cLl, cMl, cRl, cLr, cMr, cRr, ":", pos, val);

			res += pos * val * cR;
		}
	}

	//dump("a2 より大，移動 :");
	repi(i, 1, n) repi(j, i + 1, n) {
		repi(cLl, 0, cL) {
			int cMl = 0;
			int cRl = (i - 1) - cLl - cMl;
			int cLr = cRl;
			int cMr = 0;
			int cRr = (n - j) - cLr - cMr;
			int cLm = (cL - 1) - cLl - cLr;
			int cMm = cM - cMl - cMr;
			int cRm = (cR - 1) - cRl - cRr;

			if (cRl < 0 || cRr < 0 || cLm < 0 || cMm < 0 || cRm < 0) continue;

			mint pos = fm.mul({ cLl, cMl, cRl }) * fm.mul({ cLm, cMm, cRm }) * fm.mul({ cLr, cMr, cRr });
			mint val = fm.fact(cL - 1) * fm.fact(cM) * fm.fact(cR - 1);
			//dump("i:", i, "j:", j, "c:", cLl, cMl, cRl, cLm, cMm, cRm, cLr, cMr, cRr, ":", pos, val);

			res += pos * val * cL * cR;
		}
	}

	//dump("a1 より大，a2 以下，不動 :");
	repi(i, 1, n) {
		repi(cLl, 0, cL) repi(cMl, 0, cM - 1) {
			int cLr = cL - cLl;
			int cMr = (cM - 1) - cMl;
			int cRl = (i - 1) - cLl - cMl;
			int cRr = (n - i) - cLr - cMr;
			if (cRl < 0 || cRr < 0) continue;
			if (cMl + cRl < cLr) continue;
			if (cRl > cLr + cMr) continue;

			mint pos = fm.mul({ cLl, cMl, cRl }) * fm.mul({ cLr, cMr, cRr });
			mint val = fm.fact(cL) * fm.fact(cM - 1) * fm.fact(cR);
			//dump("i:", i, "c:", cLl, cMl, cRl, cLr, cMr, cRr, ":", pos, val);

			res += pos * val * cM;
		}
	}

	{
		swap(a1, a2);
		a1 = n - a1;
		a2 = n - a2;
		//dump(a1, a2);

		cL = a1;
		cM = a2 - a1;
		cR = n - a2;

		//dump("a2 より大，不動 :");
		repi(i, 1, n) {
			repi(cLl, 0, cL) repi(cMl, 0, cM) {
				int cLr = cL - cLl;
				int cMr = cM - cMl;
				int cRl = (i - 1) - cLl - cMl;
				int cRr = (n - i) - cLr - cMr;
				if (cRl < 0 || cRr < 0) continue;
				if (cMl + cRl < cLr) continue;
				if (cRl < cLr + cMr) continue;

				mint pos = fm.mul({ cLl, cMl, cRl }) * fm.mul({ cLr, cMr, cRr });
				mint val = fm.fact(cL) * fm.fact(cM) * fm.fact(cR - 1);
				//dump("i:", i, "c:", cLl, cMl, cRl, cLr, cMr, cRr, ":", pos, val);

				res += pos * val * cR;
			}
		}

		//dump("a2 より大，移動 :");
		repi(i, 1, n) repi(j, i + 1, n) {
			repi(cLl, 0, cL) {
				int cMl = 0;
				int cRl = (i - 1) - cLl - cMl;
				int cLr = cRl;
				int cMr = 0;
				int cRr = (n - j) - cLr - cMr;
				int cLm = (cL - 1) - cLl - cLr;
				int cMm = cM - cMl - cMr;
				int cRm = (cR - 1) - cRl - cRr;

				if (cRl < 0 || cRr < 0 || cLm < 0 || cMm < 0 || cRm < 0) continue;

				mint pos = fm.mul({ cLl, cMl, cRl }) * fm.mul({ cLm, cMm, cRm }) * fm.mul({ cLr, cMr, cRr });
				mint val = fm.fact(cL - 1) * fm.fact(cM) * fm.fact(cR - 1);
				//dump("i:", i, "j:", j, "c:", cLl, cMl, cRl, cLm, cMm, cRm, cLr, cMr, cRr, ":", pos, val);

				res += pos * val * cL * cR;
			}
		}
	}

	return res;
}


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

	return seq;
}


// i1 = n1 に対する愚直解を返す．
vvm naive_1(int n1) {
	vvm tbl;

	return tbl;
}


// (i1,i2,i3)∈[0..n1)×[0..n2)×[0..n3) に対する愚直解を返す．
vvvm naive() {
	int N1 = 14, N2 = 14, N3 = 14;
	vvvm box;
	rep(i1, N1) {
		vvm tbl;
		rep(i2, N2) {
			vm seq;
			rep(i3, N3) {
				seq.push_back(naive_123(i1, i2, i3));
			}
			tbl.push_back(seq);
		}
		box.push_back(tbl);
	}

	//vvvm box(N1, vvm(N2, vm(N3)));
	//rep(i1, N1) {
	//	dump("i1:", i1);
	//	rep(i2, N2) rep(i3, N3) {
	//		box[i1][i2][i3] = naive_123(i1, i2, i3);
	//	}
	//}

	//vvvm box(N1, vvm(N2));
	//rep(i1, N1) {
	//	dump("i1:", i1);
	//	rep(i2, N2) {
	//		box[i1][i2] = naive_12(i1, i2);
	//	}
	//}

	//vvvm box(N1);
	//rep(i1, N1) {
	//	dump("i1:", i1);
	//	box[i1] = naive_1(i1);
	//}

#ifdef _MSC_VER
	// 埋め込み用
	string eb;
	eb += "vvvm box = {\n";
	rep(i1, sz(box)) {
		eb += "{";
		rep(i2, sz(box[i1])) {
			eb += "{";
			rep(i3, sz(box[i1][i2])) {
				eb += to_string(box[i1][i2][i3].val()) + ",";
			}
			if (eb.back() == ',') eb.pop_back();
			eb += "},";
		}
		if (eb.back() == ',') eb.pop_back();
		eb += "},\n";
	}
	eb.pop_back(); eb.pop_back();
	eb += "};\n\n";
	cout << eb;
#endif

	return box;
}


//【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);
	while (fgets(buf, sizeof(buf), fp))
	_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();
	if (v > mint::mod() / 2) v -= mint::mod();
	return to_string(v);
}


pair<vm, int> slice_1D(const vvm& tbl, int i2) {
	// seq : tbl[..][i2] を抜き出した列
	vm seq;

	// offset : seq[0] が tbl[offset][i2] に対応することを表す．
	int offset = INF;

	rep(i1, sz(tbl)) {
		if (i2 < sz(tbl[i1])) {
			chmin(offset, i1);
			seq.push_back(tbl[i1][i2]);
		}
	}

	return { seq, offset };
}


tuple<vvm, int, int> slice_2D(const vvvm& box, int i3) {
	// tbl : box[..][..][i3] を抜き出したテーブル
	vvm tbl;

	// offset : tbl[0][0] が box[offset1][offset2][i3] に対応することを表す．
	int offset1 = INF, offset2 = INF;

	rep(i1, sz(box)) {
		tbl.push_back(vm());

		rep(i2, sz(box[i1])) {
			if (i2 < sz(box[i1]) && i3 < sz(box[i1][i2])) {
				chmin(offset1, i1);
				chmin(offset2, i2);
				tbl.back().push_back(box[i1][i2][i3]);
			}
		}

		if (tbl.back().empty()) tbl.pop_back();
	}

	return { tbl, offset1, offset2 };
}


// 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, int LLL = 0) {
	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] = 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()) {
			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());
			cmd += "*(i1-";
			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．
pair<vvvm, vvvm> embed_coefs_2D(const vvm& tbl, int DEG1_ini = 1, int TRM2_ini = 1, int DEG2_ini = 1, int LLL = 0) {
	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;
						t1 = TRM1;
						d1 = DEG1;
						t2 = TRM2;
						d2 = DEG2;
						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 == 0) {
			rep(d1, DEG1) rep(t2, TRM2) rep(d2, DEG2) {
				int idx = (d1 * TRM2 + t2) * DEG2 + d2;
				coefs[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(d1, DEG1) rep(t2, TRM2) rep(d2, DEG2) {
				int idx = (d1 * TRM2 + t2) * DEG2 + d2;
				coefs[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(d1, DEG1) rep(t2, TRM2) rep(d2, DEG2) {
				int idx = (d1 * TRM2 + t2) * DEG2 + d2;
				coefs[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[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]\"";
		//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 -------");
		vvvm coefs1(TRM2 - 1);
		rep(i2, TRM2 - 1) {
			dump("--- i2:", i2, "---");

			auto [seq, offset] = slice_1D(tbl, i2);
			coefs1[i2] = embed_coefs_1D(seq, 1, 1, LLL);
		}

		return { coefs1, coefs };
	}

	return pair<vvvm, vvvm>();
}


// 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<vvvvm, vvvvm, vvvvm> embed_coefs_3D(const vvvm& box, int DEG1_ini = 1, int DEG2_ini = 1, int TRM3_ini = 1, int DEG3_ini = 1, int LLL = 0) {
	int n1 = sz(box);

	// 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) Σt3∈[0..TRM3) Σd3∈[0..DEG3)
	//			c[t1][d1][t2][d2]][t3][d3] (i1-TRM1+1+t1)^d1 (i2-TRM2+1+t2)^d2 (i3-TRM3+1+t3)^d3 box[i1-t1][i2-t2][i3-t3] = 0
	// を探す．
	int TRM1 = 1, DEG1 = DEG1_ini;
	int TRM2 = 1, DEG2 = DEG2_ini;
	int TRM3 = TRM3_ini, DEG3 = DEG3_ini;
	int P_MAX = max({ TRM1, DEG1, TRM2, DEG2, TRM3, DEG3 });

	while (1) {
		//dump("TRM1:", TRM1, "DEG1:", DEG1, "TRM2:", TRM2, "DEG2:", DEG2, "TRM3:", TRM3, "DEG3:", DEG3);

		int w = TRM1 * DEG1 * TRM2 * DEG2 * TRM3 * DEG3;

		// 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(box[i1]);
			repi(i2, TRM2 - 1, n2 - 1) {
				int n3 = sz(box[i1][i2]);
				repi(i3, TRM3 - 1, n3 - 1) {
					vm a(w); bool valid = true;
					rep(t1, TRM1) rep(d1, DEG1) rep(t2, TRM2) rep(d2, DEG2) rep(t3, TRM3) rep(d3, DEG3) {
						if (i2 - t2 >= sz(box[i1 - t1]) || i3 - t3 >= sz(box[i1 - t1][i2 - t2])) {
							valid = false;
							t1 = TRM1;
							d1 = DEG1;
							t2 = TRM2;
							d2 = DEG2;
							t3 = TRM3;
							d3 = DEG3;
							break;
						}
						int idx = ((((t1 * DEG1 + d1) * TRM2 + t2) * DEG2 + d2) * TRM3 + t3) * DEG3 + d3;
						mint pow_i = mint(i1 - TRM1 + 1 + t1).pow(d1) * mint(i2 - TRM2 + 1 + t2).pow(d2) * mint(i3 - TRM3 + 1 + t3).pow(d3);
						a[idx] = pow_i * box[i1 - t1][i2 - t2][i3 - t3];
					}
					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) {
				DEG3++;
				if (DEG3 > P_MAX) { DEG3 = 1; TRM3++; };
				if (TRM3 > P_MAX) { TRM3 = 1; DEG2++; };
				if (DEG2 > P_MAX) { DEG2 = 1; TRM2++; };
				if (TRM2 > 1)     { TRM2 = 1; DEG1++; };
				if (DEG1 > P_MAX) { DEG1 = 1; TRM1++; };
				if (TRM1 > 1)     { TRM1 = 1; P_MAX++; };
				if (max({ TRM1, DEG1, TRM2, DEG2, TRM3, DEG3 }) == P_MAX) break;
			}
			continue;
		}

		dump("TRM1:", TRM1, "DEG1:", DEG1, "TRM2:", TRM2, "DEG2:", DEG2, "TRM3:", TRM3, "DEG3:", DEG3);
		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 coefs3(DEG1, vvvm(DEG2, vvm(TRM3, vm(DEG3))));

		if (LLL == 0) {
			rep(d1, DEG1) rep(d2, DEG2) rep(t3, TRM3) rep(d3, DEG3) {
				int idx = ((d1 * DEG2 + d2) * TRM3 + t3) * DEG3 + d3;
				coefs3[d1][d2][t3][d3] = 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(d1, DEG1) rep(d2, DEG2) rep(t3, TRM3) rep(d3, DEG3) {
				int idx = ((d1 * DEG2 + d2) * TRM3 + t3) * DEG3 + d3;
				coefs3[d1][d2][t3][d3] = 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(d1, DEG1) rep(d2, DEG2) rep(t3, TRM3) rep(d3, DEG3) {
				int idx = ((d1 * DEG2 + d2) * TRM3 + t3) * DEG3 + d3;
				coefs3[d1][d2][t3][d3] = lat0[idx];
			}
		}

		// 分母チェック
#ifdef _MSC_VER
		cout << "dnm 3D:" << 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(d3, DEG3) {
			rep(d2, DEG2) {
				rep(d1, DEG1) {
					cmd += to_signed_string(coefs3[d1][d2][0][d3]);
					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 += "*(i3-";
					cmd += to_string(TRM3);
					cmd += "+1)^";
					cmd += to_string(d3);
					cmd += "+";
				}
			}
		}
		cmd.pop_back();
		cmd += "]/.x_Integer:>toFrac[x]\"";
		//dump("cmd:", cmd);

		FILE* fp = _popen(cmd.c_str(), "r");
		char buf[4096];
		while (fgets(buf, sizeof(buf), fp)) printf("%s", buf);
		_pclose(fp);
#endif

		// (i1,i2) 平面における初項の延長
		dump("------- embed_coefs_2D -------");
		vvvvm coefs1(TRM3 - 1); vvvvm coefs2(TRM3 - 1);
		rep(i3, TRM3 - 1) {
			dump("--- i3:", i3, "---");

			auto [tbl, offset1, offset2] = slice_2D(box, i3);
			tie(coefs1[i3], coefs2[i3]) = embed_coefs_2D(tbl, 1, 1, LLL);
		}

#ifdef _MSC_VER
		// 埋め込み用の文字列を出力する．
		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";
		eb += "constexpr int TRM3 = ";
		eb += to_string(TRM3);
		eb += ";\n";
		eb += "constexpr int DEG3 = ";
		eb += to_string(DEG3);
		eb += ";\n\n";

		eb += "vvvm coefs1[TRM3 - 1] = {";
		rep(i3, TRM3 - 1) {
			eb += "{\n";
			rep(i2, sz(coefs1[i3])) {
				eb += "{";
				rep(t1, sz(coefs1[i3][i2])) {
					eb += "{";
					rep(d1, sz(coefs1[i3][i2][t1])) {
						eb += to_signed_string(coefs1[i3][i2][t1][d1]) + ",";
					}
					eb.pop_back();
					eb += "},";
				}
				eb.pop_back();
				eb += "},\n";
			}
			eb.pop_back(); eb.pop_back();
			eb += "},";
		}
		eb.pop_back();
		eb += "};\n\n";

		eb += "vvvm coefs2[TRM3 - 1] = {";
		rep(i3, TRM3 - 1) {
			eb += "{\n";
			rep(d1, sz(coefs2[i3])) {
				eb += "{";
				rep(t2, sz(coefs2[i3][d1])) {
					eb += "{";
					rep(d2, sz(coefs2[i3][d1][t2])) {
						eb += to_signed_string(coefs2[i3][d1][t2][d2]) + ",";
					}
					eb.pop_back();
					eb += "},";
				}
				eb.pop_back();
				eb += "},\n";
			}
			eb.pop_back(); eb.pop_back();
			eb += "},";
		}
		eb.pop_back();
		eb += "};\n\n";

		eb += "mint coefs3[DEG1][DEG2][TRM3][DEG3] = {";
		rep(d1, DEG1) {
			eb += "{\n";
			rep(d2, DEG2) {
				eb += "{";
				rep(t3, TRM3) {
					eb += "{";
					rep(d3, DEG3) {
						eb += to_signed_string(coefs3[d1][d2][t3][d3]) + ",";
					}
					eb.pop_back();
					eb += "},";
				}
				eb.pop_back();
				eb += "},\n";
			}
			eb.pop_back(); eb.pop_back();
			eb += "},";
		}
		eb.pop_back();
		eb += "};\n\n";

		cout << eb;
#endif

		return { coefs1, coefs2, coefs3 };
	}

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


// 数列 seq を延長して seq[0..N] にする．
void solve_1D(vm& seq, int N, vvm coefs) {
	// static int call_cnt = 0;
	
	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 i1 =", i);
			Assert(dnm != 0);
		}
		seq[i] = -num / dnm;
		
		// 除算回避用
		//mint dnm_inv;
		//if (call_cnt == 0) {
		//	dnm_inv = fm.inv_neg(1 + i);
		//}
		//else {
		//	dnm_inv = fm.inv_neg(-1 + 2 * i) * 2 * fm.inv_neg(2 + i);
		//}
		//seq[i] = -num * dnm_inv;
	}
	
	// call_cnt++;
}


// 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) {
	// static int call_cnt = 0;

	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, "---");

		auto [seq, offset] = slice_1D(tbl, i2);
		if (N - offset < 0) continue;
		solve_1D(seq, N - offset, coefs1[i2]);
		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 N1 =", N, "i2 =", i2);
			Assert(dnm != 0);
		}
		res[i2] = -num / dnm;
	}

	// call_cnt++;

	return res;
}


// 3 次元数列 box を元に seq = tbl[N1][N2][0..N3] を計算する．
vm solve_3D(const vvvm& box, int N1, int N2, int N3, vvvvm coefs1, vvvvm coefs2, vvvvm coefs3) {
	int TRM1 = 1;
	int DEG1 = sz(coefs3);
	int TRM2 = 1;
	int DEG2 = sz(coefs3[0]);
	int TRM3 = sz(coefs3[0][0]);
	int DEG3 = sz(coefs3[0][0][0]);

	vm res(TRM3 - 1);

	// (i1,i2) 平面における延長
	dump("------- solve_2D -------");
	rep(i3, TRM3 - 1) {
		dump("--- i3:", i3, "---");

		auto [tbl, offset1, offset2] = slice_2D(box, i3);
		if (N1 - offset1 < 0 || N2 - offset2 < 0) continue;
		auto seq = solve_2D(tbl, N1 - offset1, N2 - offset2, coefs1[i3], coefs2[i3]);
		dump("seq:", seq);
		res[i3] = seq[N2 - offset2];
	}

	vm pow_i1s(DEG1);
	pow_i1s[0] = 1;
	repi(d1, 1, DEG1 - 1) pow_i1s[d1] = pow_i1s[d1 - 1] * (N1 - TRM1 + 1);

	vm pow_i2s(DEG2);
	pow_i2s[0] = 1;
	repi(d2, 1, DEG2 - 1) pow_i2s[d2] = pow_i2s[d2 - 1] * (N2 - TRM2 + 1);

	// i3 方向に tbl[N1][N2][..] を延長する．
	res.resize(N3 + 1);
	repi(i3, TRM3 - 1, N3) {
		mint dnm = 0;
		mint pow_i3 = 1;
		rep(d3, DEG3) {
			rep(d2, DEG2) {
				rep(d1, DEG1) {
					dnm += coefs3[d1][d2][0][d3] * pow_i1s[d1] * pow_i2s[d2] * pow_i3;
				}				
			}
			pow_i3 *= i3 - TRM3 + 1;
		}

		mint num = 0;
		repi(t3, 1, TRM3 - 1) {
			mint pow_i3 = 1;
			rep(d3, DEG3) {
				rep(d2, DEG2) {
					rep(d1, DEG1) {
						num += coefs3[d1][d2][t3][d3] * pow_i1s[d1] * pow_i2s[d2] * pow_i3 * res[i3 - t3];
					}
				}
				pow_i3 *= i3 - TRM3 + 1 + t3;
			}
		}

		// dnm * tbl[N1][N2][i3] + num = 0 を解く．分母 0 に注意！
		if (dnm == 0) {
			dump("DIVISION BY ZERO at N1 =", N1, "N2 =", N2, "i3 =", i3);
			Assert(dnm != 0);
		}
		res[i3] = -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_coefs() を実行する．
	// 3. 出力を solve() 内に貼る．
	// 4. solve(tbl, n, m) 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 box = 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, DEG2_ini, TRM3_ini, DEG3_ini, LLL
	auto [coefs1, coefs2, coefs3] = embed_coefs_3D(box, 1, 1, 1, 1, 0);

	int n, a1, a2;
	cin >> n >> a1 >> a2;
	
	if (a1 == a2) {
		mint res = n;
		repi(i, 1, n) res *= i;
		EXIT(res);
	}

	if (a1 > a2) swap(a1, a2);

	int n0 = a1 - 1, n1 = a2 - a1 - 1, n2 = n - a2 - 1;

	// 3 次元数列 box を元に seq = box[n1][n2][0..n3] 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_3D(box, n0, n1, n2, coefs1, coefs2, coefs3);
//	auto seq = solve_3D(box, n0, n1, n2);
//	dump(seq);
	
	mint res = seq[n2];
	
	EXIT(res);
}