Problem:
Score : 800 points

Problem Statement

There are N children, numbered 1,2,\ldots,N.
In the next K days, we will give them some cookies.
In the i-th day, we will choose a_i children among the N with equal probability, and give one cookie to each child chosen. (We make these K choices independently.)

Let us define the happiness of the children as c_1 \times c_2 \times \ldots \times c_N, where c_i is the number of cookies received by Child i in the K days.
Find the expected happiness multiplied by \binom{N}{a_1} \times \binom{N}{a_2} \times \ldots \times \binom{N}{a_K} (we can show that this value is an integer), modulo (10^{9}+7).

Notes

\binom{n}{k} denotes the number of possible choices of k objects out of given distinct n objects, disregarding order.

Constraints

1 \leq N \leq 1000

1 \leq K \leq 20

1 \leq a_i \leq N

Input

Input is given from Standard Input in the following format:

N K
a_1 a_2 \ldots a_K

Output

Print the answer.

Sample Input 1

3 2
3 2

Sample Output 1

12

On the first day, all the children 1, 2, and 3 receive one cookie.

On the second day, two out of the three children 1, 2, and 3 receive one cookie.

Their happiness is 4 in any case, so the expected happiness is 4. Print this value multiplied by \binom{3}{3} \times \binom{3}{2}, which is 12.

Sample Input 2

856 16
399 263 665 432 206 61 784 548 422 313 848 478 827 26 398 63

Sample Output 2

337587117

Find the expected value multiplied by \binom{N}{a_1} \times \binom{N}{a_2} \times \ldots \times \binom{N}{a_K}, modulo (10^9+7).

Solution in C++ (GCC 9.2.1) (judge verdict: Accepted)

#define  _CRT_SECURE_NO_WARNINGS
#define _USE_MATH_DEFINES
#define _SILENCE_ALL_CXX17_DEPRECATION_WARNINGS
#pragma GCC target("avx2")
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")
#pragma comment (linker, "/STACK:526000000")
#include "bits/stdc++.h"

/*
AtCoder Compressed STL, licensed under public domain (cc0)
Download Compressed: https://gist.github.com/yosupo06/ddd51afb727600fd95d9d8ad6c3c80c9
based on AtCoder STL,
Download: https://img.atcoder.jp/practice2/ac-library.zip
*/


#include <algorithm>
#include <array>

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

namespace atcoder {

	namespace internal {

		int ceil_pow2(int n) {
			int x = 0;
			while ((1U << x) < (unsigned int)(n)) x++;
			return x;
		}

		int bsf(unsigned int n) {
#ifdef _MSC_VER
			unsigned long index;
			_BitScanForward(&index, n);
			return index;
#else
			return __builtin_ctz(n);
#endif
		}

	}  // namespace internal

}  // namespace atcoder



#include <utility>

namespace atcoder {

	namespace internal {

		constexpr long long safe_mod(long long x, long long m) {
			x %= m;
			if (x < 0) x += m;
			return x;
		}

		struct barrett {
			unsigned int _m;
			unsigned long long im;

			barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}

			unsigned int umod() const { return _m; }

			unsigned int mul(unsigned int a, unsigned int b) const {

				unsigned long long z = a;
				z *= b;
#ifdef _MSC_VER
				unsigned long long x;
				_umul128(z, im, &x);
#else
				unsigned long long x =
					(unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
				unsigned int v = (unsigned int)(z - x * _m);
				if (_m <= v) v += _m;
				return v;
			}
		};

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

		constexpr bool is_prime_constexpr(int n) {
			if (n <= 1) return false;
			if (n == 2 || n == 7 || n == 61) return true;
			if (n % 2 == 0) return false;
			long long d = n - 1;
			while (d % 2 == 0) d /= 2;
			int v[3] = { 2,7,61 };
			for (long long a : v) {
				long long t = d;
				long long y = pow_mod_constexpr(a, t, n);
				while (t != n - 1 && y != 1 && y != n - 1) {
					y = y * y % n;
					t <<= 1;
				}
				if (y != n - 1 && t % 2 == 0) {
					return false;
				}
			}
			return true;
		}
		template <int n> constexpr bool is_prime = is_prime_constexpr(n);

		constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
			a = safe_mod(a, b);
			if (a == 0) return { b, 0 };

			long long s = b, t = a;
			long long m0 = 0, m1 = 1;

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


				auto tmp = s;
				s = t;
				t = tmp;
				tmp = m0;
				m0 = m1;
				m1 = tmp;
			}
			if (m0 < 0) m0 += b / s;
			return { s, m0 };
		}

		constexpr int primitive_root_constexpr(int m) {
			if (m == 2) return 1;
			if (m == 167772161) return 3;
			if (m == 469762049) return 3;
			if (m == 754974721) return 11;
			if (m == 998244353) return 3;
			int divs[20] = {};
			divs[0] = 2;
			int cnt = 1;
			int x = (m - 1) / 2;
			while (x % 2 == 0) x /= 2;
			for (int i = 3; (long long)(i)*i <= x; i += 2) {
				if (x % i == 0) {
					divs[cnt++] = i;
					while (x % i == 0) {
						x /= i;
					}
				}
			}
			if (x > 1) {
				divs[cnt++] = x;
			}
			for (int g = 2;; g++) {
				bool ok = true;
				for (int i = 0; i < cnt; i++) {
					if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
						ok = false;
						break;
					}
				}
				if (ok) return g;
			}
		}
		template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

	}  // namespace internal

}  // namespace atcoder


#include <cassert>
#include <numeric>
#include <type_traits>

namespace atcoder {

	namespace internal {

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

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

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

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

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

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

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

#else

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

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

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

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

#endif

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

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

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

	}  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <numeric>
#include <type_traits>

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

namespace atcoder {

	namespace internal {

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

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

	}  // namespace internal

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

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

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

		unsigned int val() const { return _v; }

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

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

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

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

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

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

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

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

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

		unsigned int val() const { return _v; }

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

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

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

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

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

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

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

	namespace internal {

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

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

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

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

	}  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <type_traits>
#include <vector>

namespace atcoder {

	namespace internal {

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

			static bool first = true;
			static mint sum_e[30];  // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]
			if (first) {
				first = false;
				mint es[30], ies[30];  // es[i]^(2^(2+i)) == 1
				int cnt2 = bsf(mint::mod() - 1);
				mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
				for (int i = cnt2; i >= 2; i--) {
					es[i - 2] = e;
					ies[i - 2] = ie;
					e *= e;
					ie *= ie;
				}
				mint now = 1;
				for (int i = 0; i < cnt2 - 2; i++) {
					sum_e[i] = es[i] * now;
					now *= ies[i];
				}
			}
			for (int ph = 1; ph <= h; ph++) {
				int w = 1 << (ph - 1), p = 1 << (h - ph);
				mint now = 1;
				for (int s = 0; s < w; s++) {
					int offset = s << (h - ph + 1);
					for (int i = 0; i < p; i++) {
						auto l = a[i + offset];
						auto r = a[i + offset + p] * now;
						a[i + offset] = l + r;
						a[i + offset + p] = l - r;
					}
					now *= sum_e[bsf(~(unsigned int)(s))];
				}
			}
		}

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

			static bool first = true;
			static mint sum_ie[30];  // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]
			if (first) {
				first = false;
				mint es[30], ies[30];  // es[i]^(2^(2+i)) == 1
				int cnt2 = bsf(mint::mod() - 1);
				mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
				for (int i = cnt2; i >= 2; i--) {
					es[i - 2] = e;
					ies[i - 2] = ie;
					e *= e;
					ie *= ie;
				}
				mint now = 1;
				for (int i = 0; i < cnt2 - 2; i++) {
					sum_ie[i] = ies[i] * now;
					now *= es[i];
				}
			}

			for (int ph = h; ph >= 1; ph--) {
				int w = 1 << (ph - 1), p = 1 << (h - ph);
				mint inow = 1;
				for (int s = 0; s < w; s++) {
					int offset = s << (h - ph + 1);
					for (int i = 0; i < p; i++) {
						auto l = a[i + offset];
						auto r = a[i + offset + p];
						a[i + offset] = l + r;
						a[i + offset + p] =
							(unsigned long long)(mint::mod() + l.val() - r.val()) *
							inow.val();
					}
					inow *= sum_ie[bsf(~(unsigned int)(s))];
				}
			}
		}

	}  // namespace internal

	template <class mint, internal::is_static_modint_t<mint>* = nullptr>
	std::vector<mint> convolution(std::vector<mint> a, std::vector<mint> b) {
		int n = int(a.size()), m = int(b.size());
		if (!n || !m) return {};
		if (std::min(n, m) <= 60) {
			if (n < m) {
				std::swap(n, m);
				std::swap(a, b);
			}
			std::vector<mint> ans(n + m - 1);
			for (int i = 0; i < n; i++) {
				for (int j = 0; j < m; j++) {
					ans[i + j] += a[i] * b[j];
				}
			}
			return ans;
		}
		int z = 1 << internal::ceil_pow2(n + m - 1);
		a.resize(z);
		internal::butterfly(a);
		b.resize(z);
		internal::butterfly(b);
		for (int i = 0; i < z; i++) {
			a[i] *= b[i];
		}
		internal::butterfly_inv(a);
		a.resize(n + m - 1);
		mint iz = mint(z).inv();
		for (int i = 0; i < n + m - 1; i++) a[i] *= iz;
		return a;
	}

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

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

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

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

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

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

		std::vector<long long> c(n + m - 1);
		for (int i = 0; i < n + m - 1; i++) {
			unsigned long long x = 0;
			x += (c1[i] * i1) % MOD1 * M2M3;
			x += (c2[i] * i2) % MOD2 * M1M3;
			x += (c3[i] * i3) % MOD3 * M1M2;
			long long diff =
				c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));
			if (diff < 0) diff += MOD1;
			static constexpr unsigned long long offset[5] = {
				0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3 };
			x -= offset[diff % 5];
			c[i] = x;
		}

		return c;
	}

}  // namespace atcoder


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

namespace atcoder {

	struct dsu {
	public:
		dsu() : _n(0) {}
		dsu(int n) : _n(n), parent_or_size(n, -1) {}

		int merge(int a, int b) {
			assert(0 <= a && a < _n);
			assert(0 <= b && b < _n);
			int x = leader(a), y = leader(b);
			if (x == y) return x;
			if (-parent_or_size[x] < -parent_or_size[y]) std::swap(x, y);
			parent_or_size[x] += parent_or_size[y];
			parent_or_size[y] = x;
			return x;
		}

		bool same(int a, int b) {
			assert(0 <= a && a < _n);
			assert(0 <= b && b < _n);
			return leader(a) == leader(b);
		}

		int leader(int a) {
			assert(0 <= a && a < _n);
			if (parent_or_size[a] < 0) return a;
			return parent_or_size[a] = leader(parent_or_size[a]);
		}

		int size(int a) {
			assert(0 <= a && a < _n);
			return -parent_or_size[leader(a)];
		}

		std::vector<std::vector<int>> groups() {
			std::vector<int> leader_buf(_n), group_size(_n);
			for (int i = 0; i < _n; i++) {
				leader_buf[i] = leader(i);
				group_size[leader_buf[i]]++;
			}
			std::vector<std::vector<int>> result(_n);
			for (int i = 0; i < _n; i++) {
				result[i].reserve(group_size[i]);
			}
			for (int i = 0; i < _n; i++) {
				result[leader_buf[i]].push_back(i);
			}
			result.erase(
				std::remove_if(result.begin(), result.end(),
					[&](const std::vector<int>& v) { return v.empty(); }),
				result.end());
			return result;
		}

	private:
		int _n;
		std::vector<int> parent_or_size;
	};

}  // namespace atcoder


#include <cassert>
#include <vector>

namespace atcoder {

	template <class T> struct fenwick_tree {
		using U = internal::to_unsigned_t<T>;

	public:
		fenwick_tree() : _n(0) {}
		fenwick_tree(int n) : _n(n), data(n) {}

		void add(int p, T x) {
			assert(0 <= p && p < _n);
			p++;
			while (p <= _n) {
				data[p - 1] += U(x);
				p += p & -p;
			}
		}

		T sum(int l, int r) {
			assert(0 <= l && l <= r && r <= _n);
			return sum(r) - sum(l);
		}

	private:
		int _n;
		std::vector<U> data;

		U sum(int r) {
			U s = 0;
			while (r > 0) {
				s += data[r - 1];
				r -= r & -r;
			}
			return s;
		}
	};

}  // namespace atcoder


#include <algorithm>
#include <cassert>
#include <iostream>
#include <vector>
namespace atcoder {

	template <class S,
		S(*op)(S, S),
		S(*e)(),
		class F,
		S(*mapping)(F, S),
		F(*composition)(F, F),
		F(*id)()>
		struct lazy_segtree {
		public:
			lazy_segtree() : lazy_segtree(0) {}
			lazy_segtree(int n) : lazy_segtree(std::vector<S>(n, e())) {}
			lazy_segtree(const std::vector<S>& v) : _n(int(v.size())) {
				log = internal::ceil_pow2(_n);
				size = 1 << log;
				d = std::vector<S>(2 * size, e());
				lz = std::vector<F>(size, id());
				for (int i = 0; i < _n; i++) d[size + i] = v[i];
				for (int i = size - 1; i >= 1; i--) {
					update(i);
				}
			}

			void set(int p, S x) {
				assert(0 <= p && p < _n);
				p += size;
				for (int i = log; i >= 1; i--) push(p >> i);
				d[p] = x;
				for (int i = 1; i <= log; i++) update(p >> i);
			}

			S get(int p) {
				assert(0 <= p && p < _n);
				p += size;
				for (int i = log; i >= 1; i--) push(p >> i);
				return d[p];
			}

			S prod(int l, int r) {
				assert(0 <= l && l <= r && r <= _n);
				if (l == r) return e();

				l += size;
				r += size;

				for (int i = log; i >= 1; i--) {
					if (((l >> i) << i) != l) push(l >> i);
					if (((r >> i) << i) != r) push(r >> i);
				}

				S sml = e(), smr = e();
				while (l < r) {
					if (l & 1) sml = op(sml, d[l++]);
					if (r & 1) smr = op(d[--r], smr);
					l >>= 1;
					r >>= 1;
				}

				return op(sml, smr);
			}

			S all_prod() { return d[1]; }

			void apply(int p, F f) {
				assert(0 <= p && p < _n);
				p += size;
				for (int i = log; i >= 1; i--) push(p >> i);
				d[p] = mapping(f, d[p]);
				for (int i = 1; i <= log; i++) update(p >> i);
			}
			void apply(int l, int r, F f) {
				assert(0 <= l && l <= r && r <= _n);
				if (l == r) return;

				l += size;
				r += size;

				for (int i = log; i >= 1; i--) {
					if (((l >> i) << i) != l) push(l >> i);
					if (((r >> i) << i) != r) push((r - 1) >> i);
				}

				{
					int l2 = l, r2 = r;
					while (l < r) {
						if (l & 1) all_apply(l++, f);
						if (r & 1) all_apply(--r, f);
						l >>= 1;
						r >>= 1;
					}
					l = l2;
					r = r2;
				}

				for (int i = 1; i <= log; i++) {
					if (((l >> i) << i) != l) update(l >> i);
					if (((r >> i) << i) != r) update((r - 1) >> i);
				}
			}

			template <bool (*g)(S)> int max_right(int l) {
				return max_right(l, [](S x) { return g(x); });
			}
			template <class G> int max_right(int l, G g) {
				assert(0 <= l && l <= _n);
				assert(g(e()));
				if (l == _n) return _n;
				l += size;
				for (int i = log; i >= 1; i--) push(l >> i);
				S sm = e();
				do {
					while (l % 2 == 0) l >>= 1;
					if (!g(op(sm, d[l]))) {
						while (l < size) {
							push(l);
							l = (2 * l);
							if (g(op(sm, d[l]))) {
								sm = op(sm, d[l]);
								l++;
							}
						}
						return l - size;
					}
					sm = op(sm, d[l]);
					l++;
				} while ((l & -l) != l);
				return _n;
			}

			template <bool (*g)(S)> int min_left(int r) {
				return min_left(r, [](S x) { return g(x); });
			}
			template <class G> int min_left(int r, G g) {
				assert(0 <= r && r <= _n);
				assert(g(e()));
				if (r == 0) return 0;
				r += size;
				for (int i = log; i >= 1; i--) push((r - 1) >> i);
				S sm = e();
				do {
					r--;
					while (r > 1 && (r % 2)) r >>= 1;
					if (!g(op(d[r], sm))) {
						while (r < size) {
							push(r);
							r = (2 * r + 1);
							if (g(op(d[r], sm))) {
								sm = op(d[r], sm);
								r--;
							}
						}
						return r + 1 - size;
					}
					sm = op(d[r], sm);
				} while ((r & -r) != r);
				return 0;
			}

		private:
			int _n, size, log;
			std::vector<S> d;
			std::vector<F> lz;

			void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
			void all_apply(int k, F f) {
				d[k] = mapping(f, d[k]);
				if (k < size) lz[k] = composition(f, lz[k]);
			}
			void push(int k) {
				all_apply(2 * k, lz[k]);
				all_apply(2 * k + 1, lz[k]);
				lz[k] = id();
			}
	};

}  // namespace atcoder


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

namespace atcoder {

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

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

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


			long long g, im;
			std::tie(g, im) = internal::inv_gcd(m0, m1);

			long long u1 = (m1 / g);
			if ((r1 - r0) % g) return { 0, 0 };

			long long x = (r1 - r0) / g % u1 * im % u1;

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

	long long floor_sum(long long n, long long m, long long a, long long b) {
		long long ans = 0;
		if (a >= m) {
			ans += (n - 1) * n * (a / m) / 2;
			a %= m;
		}
		if (b >= m) {
			ans += n * (b / m);
			b %= m;
		}

		long long y_max = (a * n + b) / m, x_max = (y_max * m - b);
		if (y_max == 0) return ans;
		ans += (n - (x_max + a - 1) / a) * y_max;
		ans += floor_sum(y_max, a, m, (a - x_max % a) % a);
		return ans;
	}

}  // namespace atcoder


#include <algorithm>

#include <vector>

namespace atcoder {

	namespace internal {

		template <class T> struct simple_queue {
			std::vector<T> payload;
			int pos = 0;
			void reserve(int n) { payload.reserve(n); }
			int size() const { return int(payload.size()) - pos; }
			bool empty() const { return pos == int(payload.size()); }
			void push(const T& t) { payload.push_back(t); }
			T& front() { return payload[pos]; }
			void clear() {
				payload.clear();
				pos = 0;
			}
			void pop() { pos++; }
		};

	}  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <limits>
#include <queue>
#include <vector>

namespace atcoder {

	template <class Cap> struct mf_graph {
	public:
		mf_graph() : _n(0) {}
		mf_graph(int n) : _n(n), g(n) {}

		int add_edge(int from, int to, Cap cap) {
			assert(0 <= from && from < _n);
			assert(0 <= to && to < _n);
			assert(0 <= cap);
			int m = int(pos.size());
			pos.push_back({ from, int(g[from].size()) });
			g[from].push_back(_edge{ to, int(g[to].size()), cap });
			g[to].push_back(_edge{ from, int(g[from].size()) - 1, 0 });
			return m;
		}

		struct edge {
			int from, to;
			Cap cap, flow;
		};

		edge get_edge(int i) {
			int m = int(pos.size());
			assert(0 <= i && i < m);
			auto _e = g[pos[i].first][pos[i].second];
			auto _re = g[_e.to][_e.rev];
			return edge{ pos[i].first, _e.to, _e.cap + _re.cap, _re.cap };
		}
		std::vector<edge> edges() {
			int m = int(pos.size());
			std::vector<edge> result;
			for (int i = 0; i < m; i++) {
				result.push_back(get_edge(i));
			}
			return result;
		}
		void change_edge(int i, Cap new_cap, Cap new_flow) {
			int m = int(pos.size());
			assert(0 <= i && i < m);
			assert(0 <= new_flow && new_flow <= new_cap);
			auto& _e = g[pos[i].first][pos[i].second];
			auto& _re = g[_e.to][_e.rev];
			_e.cap = new_cap - new_flow;
			_re.cap = new_flow;
		}

		Cap flow(int s, int t) {
			return flow(s, t, std::numeric_limits<Cap>::max());
		}
		Cap flow(int s, int t, Cap flow_limit) {
			assert(0 <= s && s < _n);
			assert(0 <= t && t < _n);

			std::vector<int> level(_n), iter(_n);
			internal::simple_queue<int> que;

			auto bfs = [&]() {
				std::fill(level.begin(), level.end(), -1);
				level[s] = 0;
				que.clear();
				que.push(s);
				while (!que.empty()) {
					int v = que.front();
					que.pop();
					for (auto e : g[v]) {
						if (e.cap == 0 || level[e.to] >= 0) continue;
						level[e.to] = level[v] + 1;
						if (e.to == t) return;
						que.push(e.to);
					}
				}
			};
			auto dfs = [&](auto self, int v, Cap up) {
				if (v == s) return up;
				Cap res = 0;
				int level_v = level[v];
				for (int& i = iter[v]; i < int(g[v].size()); i++) {
					_edge& e = g[v][i];
					if (level_v <= level[e.to] || g[e.to][e.rev].cap == 0) continue;
					Cap d =
						self(self, e.to, std::min(up - res, g[e.to][e.rev].cap));
					if (d <= 0) continue;
					g[v][i].cap += d;
					g[e.to][e.rev].cap -= d;
					res += d;
					if (res == up) break;
				}
				return res;
			};

			Cap flow = 0;
			while (flow < flow_limit) {
				bfs();
				if (level[t] == -1) break;
				std::fill(iter.begin(), iter.end(), 0);
				while (flow < flow_limit) {
					Cap f = dfs(dfs, t, flow_limit - flow);
					if (!f) break;
					flow += f;
				}
			}
			return flow;
		}

		std::vector<bool> min_cut(int s) {
			std::vector<bool> visited(_n);
			internal::simple_queue<int> que;
			que.push(s);
			while (!que.empty()) {
				int p = que.front();
				que.pop();
				visited[p] = true;
				for (auto e : g[p]) {
					if (e.cap && !visited[e.to]) {
						visited[e.to] = true;
						que.push(e.to);
					}
				}
			}
			return visited;
		}

	private:
		int _n;
		struct _edge {
			int to, rev;
			Cap cap;
		};
		std::vector<std::pair<int, int>> pos;
		std::vector<std::vector<_edge>> g;
	};

}  // namespace atcoder


#include <algorithm>
#include <cassert>
#include <limits>
#include <queue>
#include <vector>

namespace atcoder {

	template <class Cap, class Cost> struct mcf_graph {
	public:
		mcf_graph() {}
		mcf_graph(int n) : _n(n), g(n) {}

		int add_edge(int from, int to, Cap cap, Cost cost) {
			assert(0 <= from && from < _n);
			assert(0 <= to && to < _n);
			int m = int(pos.size());
			pos.push_back({ from, int(g[from].size()) });
			g[from].push_back(_edge{ to, int(g[to].size()), cap, cost });
			g[to].push_back(_edge{ from, int(g[from].size()) - 1, 0, -cost });
			return m;
		}

		struct edge {
			int from, to;
			Cap cap, flow;
			Cost cost;
		};

		edge get_edge(int i) {
			int m = int(pos.size());
			assert(0 <= i && i < m);
			auto _e = g[pos[i].first][pos[i].second];
			auto _re = g[_e.to][_e.rev];
			return edge{
				pos[i].first, _e.to, _e.cap + _re.cap, _re.cap, _e.cost,
			};
		}
		std::vector<edge> edges() {
			int m = int(pos.size());
			std::vector<edge> result(m);
			for (int i = 0; i < m; i++) {
				result[i] = get_edge(i);
			}
			return result;
		}

		std::pair<Cap, Cost> flow(int s, int t) {
			return flow(s, t, std::numeric_limits<Cap>::max());
		}
		std::pair<Cap, Cost> flow(int s, int t, Cap flow_limit) {
			return slope(s, t, flow_limit).back();
		}
		std::vector<std::pair<Cap, Cost>> slope(int s, int t) {
			return slope(s, t, std::numeric_limits<Cap>::max());
		}
		std::vector<std::pair<Cap, Cost>> slope(int s, int t, Cap flow_limit) {
			assert(0 <= s && s < _n);
			assert(0 <= t && t < _n);
			assert(s != t);
			std::vector<Cost> dual(_n, 0), dist(_n);
			std::vector<int> pv(_n), pe(_n);
			std::vector<bool> vis(_n);
			auto dual_ref = [&]() {
				std::fill(dist.begin(), dist.end(),
					std::numeric_limits<Cost>::max());
				std::fill(pv.begin(), pv.end(), -1);
				std::fill(pe.begin(), pe.end(), -1);
				std::fill(vis.begin(), vis.end(), false);
				struct Q {
					Cost key;
					int to;
					bool operator<(Q r) const { return key > r.key; }
				};
				std::priority_queue<Q> que;
				dist[s] = 0;
				que.push(Q{ 0, s });
				while (!que.empty()) {
					int v = que.top().to;
					que.pop();
					if (vis[v]) continue;
					vis[v] = true;
					if (v == t) break;
					for (int i = 0; i < int(g[v].size()); i++) {
						auto e = g[v][i];
						if (vis[e.to] || !e.cap) continue;
						Cost cost = e.cost - dual[e.to] + dual[v];
						if (dist[e.to] - dist[v] > cost) {
							dist[e.to] = dist[v] + cost;
							pv[e.to] = v;
							pe[e.to] = i;
							que.push(Q{ dist[e.to], e.to });
						}
					}
				}
				if (!vis[t]) {
					return false;
				}

				for (int v = 0; v < _n; v++) {
					if (!vis[v]) continue;
					dual[v] -= dist[t] - dist[v];
				}
				return true;
			};
			Cap flow = 0;
			Cost cost = 0, prev_cost = -1;
			std::vector<std::pair<Cap, Cost>> result;
			result.push_back({ flow, cost });
			while (flow < flow_limit) {
				if (!dual_ref()) break;
				Cap c = flow_limit - flow;
				for (int v = t; v != s; v = pv[v]) {
					c = std::min(c, g[pv[v]][pe[v]].cap);
				}
				for (int v = t; v != s; v = pv[v]) {
					auto& e = g[pv[v]][pe[v]];
					e.cap -= c;
					g[v][e.rev].cap += c;
				}
				Cost d = -dual[s];
				flow += c;
				cost += c * d;
				if (prev_cost == d) {
					result.pop_back();
				}
				result.push_back({ flow, cost });
				prev_cost = cost;
			}
			return result;
		}

	private:
		int _n;

		struct _edge {
			int to, rev;
			Cap cap;
			Cost cost;
		};

		std::vector<std::pair<int, int>> pos;
		std::vector<std::vector<_edge>> g;
	};

}  // namespace atcoder


#include <algorithm>

#include <algorithm>
#include <utility>
#include <vector>

namespace atcoder {
	namespace internal {

		template <class E> struct csr {
			std::vector<int> start;
			std::vector<E> elist;
			csr(int n, const std::vector<std::pair<int, E>>& edges)
				: start(n + 1), elist(edges.size()) {
				for (auto e : edges) {
					start[e.first + 1]++;
				}
				for (int i = 1; i <= n; i++) {
					start[i] += start[i - 1];
				}
				auto counter = start;
				for (auto e : edges) {
					elist[counter[e.first]++] = e.second;
				}
			}
		};

		struct scc_graph {
		public:
			scc_graph(int n) : _n(n) {}

			int num_vertices() { return _n; }

			void add_edge(int from, int to) { edges.push_back({ from, {to} }); }

			std::pair<int, std::vector<int>> scc_ids() {
				auto g = csr<edge>(_n, edges);
				int now_ord = 0, group_num = 0;
				std::vector<int> visited, low(_n), ord(_n, -1), ids(_n);
				visited.reserve(_n);
				auto dfs = [&](auto self, int v) -> void {
					low[v] = ord[v] = now_ord++;
					visited.push_back(v);
					for (int i = g.start[v]; i < g.start[v + 1]; i++) {
						auto to = g.elist[i].to;
						if (ord[to] == -1) {
							self(self, to);
							low[v] = std::min(low[v], low[to]);
						}
						else {
							low[v] = std::min(low[v], ord[to]);
						}
					}
					if (low[v] == ord[v]) {
						while (true) {
							int u = visited.back();
							visited.pop_back();
							ord[u] = _n;
							ids[u] = group_num;
							if (u == v) break;
						}
						group_num++;
					}
				};
				for (int i = 0; i < _n; i++) {
					if (ord[i] == -1) dfs(dfs, i);
				}
				for (auto& x : ids) {
					x = group_num - 1 - x;
				}
				return { group_num, ids };
			}

			std::vector<std::vector<int>> scc() {
				auto ids = scc_ids();
				int group_num = ids.first;
				std::vector<int> counts(group_num);
				for (auto x : ids.second) counts[x]++;
				std::vector<std::vector<int>> groups(ids.first);
				for (int i = 0; i < group_num; i++) {
					groups[i].reserve(counts[i]);
				}
				for (int i = 0; i < _n; i++) {
					groups[ids.second[i]].push_back(i);
				}
				return groups;
			}

		private:
			int _n;
			struct edge {
				int to;
			};
			std::vector<std::pair<int, edge>> edges;
		};

	}  // namespace internal

}  // namespace atcoder

#include <cassert>
#include <vector>

namespace atcoder {

	struct scc_graph {
	public:
		scc_graph() : internal(0) {}
		scc_graph(int n) : internal(n) {}

		void add_edge(int from, int to) {
			int n = internal.num_vertices();
			assert(0 <= from && from < n);
			assert(0 <= to && to < n);
			internal.add_edge(from, to);
		}

		std::vector<std::vector<int>> scc() { return internal.scc(); }

	private:
		internal::scc_graph internal;
	};

}  // namespace atcoder


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

namespace atcoder {

	template <class S, S(*op)(S, S), S(*e)()> struct segtree {
	public:
		segtree() : segtree(0) {}
		segtree(int n) : segtree(std::vector<S>(n, e())) {}
		segtree(const std::vector<S>& v) : _n(int(v.size())) {
			log = internal::ceil_pow2(_n);
			size = 1 << log;
			d = std::vector<S>(2 * size, e());
			for (int i = 0; i < _n; i++) d[size + i] = v[i];
			for (int i = size - 1; i >= 1; i--) {
				update(i);
			}
		}

		void set(int p, S x) {
			assert(0 <= p && p < _n);
			p += size;
			d[p] = x;
			for (int i = 1; i <= log; i++) update(p >> i);
		}

		S get(int p) {
			assert(0 <= p && p < _n);
			return d[p + size];
		}

		S prod(int l, int r) {
			assert(0 <= l && l <= r && r <= _n);
			S sml = e(), smr = e();
			l += size;
			r += size;

			while (l < r) {
				if (l & 1) sml = op(sml, d[l++]);
				if (r & 1) smr = op(d[--r], smr);
				l >>= 1;
				r >>= 1;
			}
			return op(sml, smr);
		}

		S all_prod() { return d[1]; }

		template <bool (*f)(S)> int max_right(int l) {
			return max_right(l, [](S x) { return f(x); });
		}
		template <class F> int max_right(int l, F f) {
			assert(0 <= l && l <= _n);
			assert(f(e()));
			if (l == _n) return _n;
			l += size;
			S sm = e();
			do {
				while (l % 2 == 0) l >>= 1;
				if (!f(op(sm, d[l]))) {
					while (l < size) {
						l = (2 * l);
						if (f(op(sm, d[l]))) {
							sm = op(sm, d[l]);
							l++;
						}
					}
					return l - size;
				}
				sm = op(sm, d[l]);
				l++;
			} while ((l & -l) != l);
			return _n;
		}

		template <bool (*f)(S)> int min_left(int r) {
			return min_left(r, [](S x) { return f(x); });
		}
		template <class F> int min_left(int r, F f) {
			assert(0 <= r && r <= _n);
			assert(f(e()));
			if (r == 0) return 0;
			r += size;
			S sm = e();
			do {
				r--;
				while (r > 1 && (r % 2)) r >>= 1;
				if (!f(op(d[r], sm))) {
					while (r < size) {
						r = (2 * r + 1);
						if (f(op(d[r], sm))) {
							sm = op(d[r], sm);
							r--;
						}
					}
					return r + 1 - size;
				}
				sm = op(d[r], sm);
			} while ((r & -r) != r);
			return 0;
		}

	private:
		int _n, size, log;
		std::vector<S> d;

		void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
	};

}  // namespace atcoder


#include <algorithm>
#include <cassert>
#include <numeric>
#include <string>
#include <vector>

namespace atcoder {

	namespace internal {

		std::vector<int> sa_naive(const std::vector<int>& s) {
			int n = int(s.size());
			std::vector<int> sa(n);
			std::iota(sa.begin(), sa.end(), 0);
			std::sort(sa.begin(), sa.end(), [&](int l, int r) {
				if (l == r) return false;
				while (l < n && r < n) {
					if (s[l] != s[r]) return s[l] < s[r];
					l++;
					r++;
				}
				return l == n;
				});
			return sa;
		}

		std::vector<int> sa_doubling(const std::vector<int>& s) {
			int n = int(s.size());
			std::vector<int> sa(n), rnk = s, tmp(n);
			std::iota(sa.begin(), sa.end(), 0);
			for (int k = 1; k < n; k *= 2) {
				auto cmp = [&](int x, int y) {
					if (rnk[x] != rnk[y]) return rnk[x] < rnk[y];
					int rx = x + k < n ? rnk[x + k] : -1;
					int ry = y + k < n ? rnk[y + k] : -1;
					return rx < ry;
				};
				std::sort(sa.begin(), sa.end(), cmp);
				tmp[sa[0]] = 0;
				for (int i = 1; i < n; i++) {
					tmp[sa[i]] = tmp[sa[i - 1]] + (cmp(sa[i - 1], sa[i]) ? 1 : 0);
				}
				std::swap(tmp, rnk);
			}
			return sa;
		}

		template <int THRESHOLD_NAIVE = 10, int THRESHOLD_DOUBLING = 40>
		std::vector<int> sa_is(const std::vector<int>& s, int upper) {
			int n = int(s.size());
			if (n == 0) return {};
			if (n == 1) return { 0 };
			if (n == 2) {
				if (s[0] < s[1]) {
					return { 0, 1 };
				}
				else {
					return { 1, 0 };
				}
			}
			if (n < THRESHOLD_NAIVE) {
				return sa_naive(s);
			}
			if (n < THRESHOLD_DOUBLING) {
				return sa_doubling(s);
			}

			std::vector<int> sa(n);
			std::vector<bool> ls(n);
			for (int i = n - 2; i >= 0; i--) {
				ls[i] = (s[i] == s[i + 1]) ? ls[i + 1] : (s[i] < s[i + 1]);
			}
			std::vector<int> sum_l(upper + 1), sum_s(upper + 1);
			for (int i = 0; i < n; i++) {
				if (!ls[i]) {
					sum_s[s[i]]++;
				}
				else {
					sum_l[s[i] + 1]++;
				}
			}
			for (int i = 0; i <= upper; i++) {
				sum_s[i] += sum_l[i];
				if (i < upper) sum_l[i + 1] += sum_s[i];
			}

			auto induce = [&](const std::vector<int>& lms) {
				std::fill(sa.begin(), sa.end(), -1);
				std::vector<int> buf(upper + 1);
				std::copy(sum_s.begin(), sum_s.end(), buf.begin());
				for (auto d : lms) {
					if (d == n) continue;
					sa[buf[s[d]]++] = d;
				}
				std::copy(sum_l.begin(), sum_l.end(), buf.begin());
				sa[buf[s[n - 1]]++] = n - 1;
				for (int i = 0; i < n; i++) {
					int v = sa[i];
					if (v >= 1 && !ls[v - 1]) {
						sa[buf[s[v - 1]]++] = v - 1;
					}
				}
				std::copy(sum_l.begin(), sum_l.end(), buf.begin());
				for (int i = n - 1; i >= 0; i--) {
					int v = sa[i];
					if (v >= 1 && ls[v - 1]) {
						sa[--buf[s[v - 1] + 1]] = v - 1;
					}
				}
			};

			std::vector<int> lms_map(n + 1, -1);
			int m = 0;
			for (int i = 1; i < n; i++) {
				if (!ls[i - 1] && ls[i]) {
					lms_map[i] = m++;
				}
			}
			std::vector<int> lms;
			lms.reserve(m);
			for (int i = 1; i < n; i++) {
				if (!ls[i - 1] && ls[i]) {
					lms.push_back(i);
				}
			}

			induce(lms);

			if (m) {
				std::vector<int> sorted_lms;
				sorted_lms.reserve(m);
				for (int v : sa) {
					if (lms_map[v] != -1) sorted_lms.push_back(v);
				}
				std::vector<int> rec_s(m);
				int rec_upper = 0;
				rec_s[lms_map[sorted_lms[0]]] = 0;
				for (int i = 1; i < m; i++) {
					int l = sorted_lms[i - 1], r = sorted_lms[i];
					int end_l = (lms_map[l] + 1 < m) ? lms[lms_map[l] + 1] : n;
					int end_r = (lms_map[r] + 1 < m) ? lms[lms_map[r] + 1] : n;
					bool same = true;
					if (end_l - l != end_r - r) {
						same = false;
					}
					else {
						while (l < end_l) {
							if (s[l] != s[r]) {
								break;
							}
							l++;
							r++;
						}
						if (l == n || s[l] != s[r]) same = false;
					}
					if (!same) rec_upper++;
					rec_s[lms_map[sorted_lms[i]]] = rec_upper;
				}

				auto rec_sa =
					sa_is<THRESHOLD_NAIVE, THRESHOLD_DOUBLING>(rec_s, rec_upper);

				for (int i = 0; i < m; i++) {
					sorted_lms[i] = lms[rec_sa[i]];
				}
				induce(sorted_lms);
			}
			return sa;
		}

	}  // namespace internal

	std::vector<int> suffix_array(const std::vector<int>& s, int upper) {
		assert(0 <= upper);
		for (int d : s) {
			assert(0 <= d && d <= upper);
		}
		auto sa = internal::sa_is(s, upper);
		return sa;
	}

	template <class T> std::vector<int> suffix_array(const std::vector<T>& s) {
		int n = int(s.size());
		std::vector<int> idx(n);
		iota(idx.begin(), idx.end(), 0);
		sort(idx.begin(), idx.end(), [&](int l, int r) { return s[l] < s[r]; });
		std::vector<int> s2(n);
		int now = 0;
		for (int i = 0; i < n; i++) {
			if (i && s[idx[i - 1]] != s[idx[i]]) now++;
			s2[idx[i]] = now;
		}
		return internal::sa_is(s2, now);
	}

	std::vector<int> suffix_array(const std::string& s) {
		int n = int(s.size());
		std::vector<int> s2(n);
		for (int i = 0; i < n; i++) {
			s2[i] = s[i];
		}
		return internal::sa_is(s2, 255);
	}

	template <class T>
	std::vector<int> lcp_array(const std::vector<T>& s,
		const std::vector<int>& sa) {
		int n = int(s.size());
		assert(n >= 1);
		std::vector<int> rnk(n);
		for (int i = 0; i < n; i++) {
			rnk[sa[i]] = i;
		}
		std::vector<int> lcp(n - 1);
		int h = 0;
		for (int i = 0; i < n; i++) {
			if (h > 0) h--;
			if (rnk[i] == 0) continue;
			int j = sa[rnk[i] - 1];
			for (; j + h < n && i + h < n; h++) {
				if (s[j + h] != s[i + h]) break;
			}
			lcp[rnk[i] - 1] = h;
		}
		return lcp;
	}

	std::vector<int> lcp_array(const std::string& s, const std::vector<int>& sa) {
		int n = int(s.size());
		std::vector<int> s2(n);
		for (int i = 0; i < n; i++) {
			s2[i] = s[i];
		}
		return lcp_array(s2, sa);
	}

	template <class T> std::vector<int> z_algorithm(const std::vector<T>& s) {
		int n = int(s.size());
		if (n == 0) return {};
		std::vector<int> z(n);
		z[0] = 0;
		for (int i = 1, j = 0; i < n; i++) {
			int& k = z[i];
			k = (j + z[j] <= i) ? 0 : std::min(j + z[j] - i, z[i - j]);
			while (i + k < n && s[k] == s[i + k]) k++;
			if (j + z[j] < i + z[i]) j = i;
		}
		z[0] = n;
		return z;
	}

	std::vector<int> z_algorithm(const std::string& s) {
		int n = int(s.size());
		std::vector<int> s2(n);
		for (int i = 0; i < n; i++) {
			s2[i] = s[i];
		}
		return z_algorithm(s2);
	}

}  // namespace atcoder


#include <cassert>
#include <vector>

namespace atcoder {

	struct two_sat {
	public:
		two_sat() : _n(0), scc(0) {}
		two_sat(int n) : _n(n), _answer(n), scc(2 * n) {}

		void add_clause(int i, bool f, int j, bool g) {
			assert(0 <= i && i < _n);
			assert(0 <= j && j < _n);
			scc.add_edge(2 * i + (f ? 0 : 1), 2 * j + (g ? 1 : 0));
			scc.add_edge(2 * j + (g ? 0 : 1), 2 * i + (f ? 1 : 0));
		}
		bool satisfiable() {
			auto id = scc.scc_ids().second;
			for (int i = 0; i < _n; i++) {
				if (id[2 * i] == id[2 * i + 1]) return false;
				_answer[i] = id[2 * i] < id[2 * i + 1];
			}
			return true;
		}
		std::vector<bool> answer() { return _answer; }

	private:
		int _n;
		std::vector<bool> _answer;
		internal::scc_graph scc;
	};

}  // namespace atcoder

/*
end of AtCoder STL
*/

#define int ll
using namespace std;
using namespace atcoder;
typedef string::const_iterator State;
#define eps 1e-11L
#define MAX_MOD 1000000007LL
#define GYAKU 500000004LL
#define MOD 998244353LL
#define pb push_back
#define mp make_pair
typedef long long ll;
typedef long double ld;
#define REP(a,b) for(long long (a) = 0;(a) < (b);++(a))
#define ALL(x) (x).begin(),(x).end()
unsigned long xor128() {
	static unsigned long x = 123456789, y = 362436069, z = 521288629, w = 88675123;
	unsigned long t = (x ^ (x << 11));
	x = y; y = z; z = w;
	return (w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)));
};
void init() {
	iostream::sync_with_stdio(false);
	cout << fixed << setprecision(20);
}

typedef complex<long double> Point;
typedef pair<complex<long double>, complex<long double>> Line;

typedef struct Circle {
	complex<long double> center;
	long double r;
}Circle;

long double dot(Point a, Point b) {
	return (a.real() * b.real() + a.imag() * b.imag());
}
long double cross(Point a, Point b) {
	return (a.real() * b.imag() - a.imag() * b.real());
}

long double Dist_Line_Point(Line a, Point b) {
	if (dot(a.second - a.first, b - a.first) < eps) return abs(b - a.first);
	if (dot(a.first - a.second, b - a.second) < eps) return abs(b - a.second);
	return abs(cross(a.second - a.first, b - a.first)) / abs(a.second - a.first);
}

int is_intersected_ls(Line a, Line b) {
	return (cross(a.second - a.first, b.first - a.first) * cross(a.second - a.first, b.second - a.first) < eps) &&
		(cross(b.second - b.first, a.first - b.first) * cross(b.second - b.first, a.second - b.first) < eps);
}

Point intersection_l(Line a, Line b) {
	Point da = a.second - a.first;
	Point db = b.second - b.first;
	return a.first + da * cross(db, b.first - a.first) / cross(db, da);
}

long double Dist_Line_Line(Line a, Line b) {
	if (is_intersected_ls(a, b) == 1) {
		return 0;
	}
	return min({ Dist_Line_Point(a,b.first), Dist_Line_Point(a,b.second),Dist_Line_Point(b,a.first),Dist_Line_Point(b,a.second) });
}

pair<Point, Point> intersection_Circle_Circle(Circle a, Circle b) {
	long double dist = abs(a.center - b.center);
	assert(dist <= eps + a.r + b.r);
	assert(dist + eps >= abs(a.r - b.r));
	Point target = b.center - a.center;
	long double pointer = target.real() * target.real() + target.imag() * target.imag();
	long double aa = pointer + a.r * a.r - b.r * b.r;
	aa /= 2.0L;
	Point l{ (aa * target.real() + target.imag() * sqrt(pointer * a.r * a.r - aa * aa)) / pointer,
			(aa * target.imag() - target.real() * sqrt(pointer * a.r * a.r - aa * aa)) / pointer };
	Point r{ (aa * target.real() - target.imag() * sqrt(pointer * a.r * a.r - aa * aa)) / pointer,
		(aa * target.imag() + target.real() * sqrt(pointer * a.r * a.r - aa * aa)) / pointer };
	r = r + a.center;
	l = l + a.center;
	return mp(l, r);
}

template< class T >
struct Matrix {
	vector< vector< T > > A;

	Matrix() {}

	Matrix(size_t n, size_t m) : A(n, vector< T >(m, 0)) {}

	Matrix(size_t n) : A(n, vector< T >(n, 0)) {};

	size_t height() const {
		return (A.size());
	}

	size_t width() const {
		return (A[0].size());
	}

	inline const vector< T >& operator[](int k) const {
		return (A.at(k));
	}

	inline vector< T >& operator[](int k) {
		return (A.at(k));
	}

	static Matrix I(size_t n) {
		Matrix mat(n);
		for (int i = 0; i < n; i++) mat[i][i] = 1;
		return (mat);
	}

	Matrix& operator+=(const Matrix& B) {
		size_t n = height(), m = width();
		assert(n == B.height() && m == B.width());
		for (int i = 0; i < n; i++)
			for (int j = 0; j < m; j++)
				(*this)[i][j] += B[i][j];
		return (*this);
	}

	Matrix& operator-=(const Matrix& B) {
		size_t n = height(), m = width();
		assert(n == B.height() && m == B.width());
		for (int i = 0; i < n; i++)
			for (int j = 0; j < m; j++)
				(*this)[i][j] -= B[i][j];
		return (*this);
	}

	Matrix& operator*=(const Matrix& B) {
		size_t n = height(), m = B.width(), p = width();
		assert(p == B.height());
		vector< vector< T > > C(n, vector< T >(m, 0));
		for (int i = 0; i < n; i++)
			for (int j = 0; j < m; j++)
				for (int k = 0; k < p; k++)
					C[i][j] = (C[i][j] + (*this)[i][k] * B[k][j]);
		A.swap(C);
		return (*this);
	}

	Matrix& operator^=(long long k) {
		Matrix B = Matrix::I(height());
		while (k > 0) {
			if (k & 1) B *= *this;
			*this *= *this;
			k >>= 1LL;
		}
		A.swap(B.A);
		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 Matrix& B) const {
		return (Matrix(*this) *= B);
	}

	Matrix operator^(const long long k) const {
		return (Matrix(*this) ^= k);
	}

	friend ostream& operator<<(ostream& os, Matrix& p) {
		size_t n = p.height(), m = p.width();
		for (int i = 0; i < n; i++) {
			os << "[";
			for (int j = 0; j < m; j++) {
				os << p[i][j] << (j + 1 == m ? "]\n" : ",");
			}
		}
		return (os);
	}


	T determinant() {
		Matrix B(*this);
		assert(width() == height());
		T ret = 1;
		for (int i = 0; i < width(); i++) {
			int idx = -1;
			for (int j = i; j < width(); j++) {
				if (B[j][i] != 0) idx = j;
			}
			if (idx == -1) return (0);
			if (i != idx) {
				ret *= -1;
				swap(B[i], B[idx]);
			}
			ret *= B[i][i];
			T vv = B[i][i];
			for (int j = 0; j < width(); j++) {
				B[i][j] /= vv;
			}
			for (int j = i + 1; j < width(); j++) {
				T a = B[j][i];
				for (int k = 0; k < width(); k++) {
					B[j][k] -= B[i][k] * a;
				}
			}
		}
		return (ret);
	}
};
#define int ll
using mint = modint1000000007;
mint mae[10000];
mint gyaku[10000];
mint dp[23][2000];
mint comb(int a, int b) {
	if (a < b) return 0;
	if (a < 0) return 0;
	mint ans = mae[a];
	ans *= gyaku[b];
	ans *= gyaku[a - b];
	return ans;
}
void solve(){
	mae[0] = 1;
	gyaku[0] = 1;
	for (int i = 1; i < 10000; ++i) {
		mae[i] = mae[i - 1] * i;
		gyaku[i] = 1 / mae[i];
	}
	int n, k;
	cin >> n >> k;
	dp[0][0] = 1;
	REP(i, k) {
		int a;
		cin >> a;
		REP(q, n+1) {
			REP(j, a + 1) {
				if (n - q < j or n - j < a - j) break;
				dp[i + 1][q + j] += dp[i][q] * comb(n - q, j) * comb(n - j, a - j);
			}
		}
	}
	cout << dp[k][n].val() << endl;
}
#undef int
int main() {
	init();
	int t = 1;
	//cin >> t;
	REP(tea, t) {
		solve();
	}
}