Problem: 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).

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.

Examples

Input

3 2
3 2


Output

12


Input

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


Output

337587117

Solution (in CPP):

Solution:
#include <bits/stdc++.h>

using namespace std;
using int64 = long long;

constexpr int DEBUG = 0;

constexpr int64 P = 1000000007;

struct FiniteField {
 private:
  int64 x;
  static inline int64 Normalize(int64 x) {
    if (0 <= x && x < P) return x; x %= P; if (x < 0) x += P; return x;
  }
 public:
  FiniteField(int64 x) : x(Normalize(x)) {}
  FiniteField() : x(0) {}
  int64 Value() const { return x; }
  inline FiniteField operator+(FiniteField o) const {
    FiniteField r(*this); r += o; return r;
  }
  inline FiniteField operator-(FiniteField o) const {
    FiniteField r(*this); r -= o; return r;
  }
  inline FiniteField operator* (FiniteField o) const {
    FiniteField r(*this); r *= o; return r;
  }
  inline FiniteField operator/ (FiniteField o) const {
    FiniteField r(*this); r /= o; return r;
  }
  inline void operator+= (FiniteField o) { x = (x + o.x) % P; }
  inline void operator-= (FiniteField o) { x = (x + P - o.x) % P; }
  inline void operator*= (FiniteField o) { x = (x * o.x) % P; }
  void operator/=(FiniteField o) {
    int64 p = P - 2; while (p) { if (p % 2) { *this *= o; } o *= o; p /= 2; }
  }
};

ostream& operator<<(ostream& s, const FiniteField& v) { s << v.Value(); return s; }

tuple<vector<FiniteField>, vector<FiniteField>> MakeFactorialTables(int n) {
  vector<FiniteField> factorials(n + 1);
  vector<FiniteField> inverse_factorials(n + 1);
  factorials[0] = 1;
  for (int i = 1; i <= n; i++) {
    factorials[i] = factorials[i - 1] * i;
  }
  inverse_factorials[n] = FiniteField(1) / factorials[n];
  for (int i = n; i >= 1; i--) {
    inverse_factorials[i - 1] = inverse_factorials[i] * i;
  }
  return make_tuple(factorials, inverse_factorials);
}

template<typename T>
vector<vector<T>> MakeVector2D(int d1, int d2, T default_value) {
  return vector<vector<T>>(d1, vector<T>(d2, default_value));
}

int main() {
  ios::sync_with_stdio(false);
  cin.tie(0);

  int n, k;
  cin >> n >> k;
  vector<int> xs(k);
  for (int i = 0; i < k; i++) {
    cin >> xs[i];
  }

  vector<FiniteField> fs, i_fs;
  tie(fs, i_fs) = MakeFactorialTables(n + 1);
  auto c_fn = [&fs, &i_fs](int a, int b) -> FiniteField {
    if (b < 0 || b > a) return 0;
    return fs[a] * i_fs[b] * i_fs[a - b];
  };

  auto dp = MakeVector2D<FiniteField>(k + 1, n + 1, 0);
  dp[0][0] = 1;
  for (int i = 0; i < k; i++) {
    int x = xs[i];
    for (int j = 0; j <= n; j++) {
      for (int a = 0; a <= n - j; a++) {
        dp[i + 1][j + a] += dp[i][j] * c_fn(n - j, a) * c_fn(x, a) / c_fn(n, a);
      }
    }
  }
  FiniteField ans = dp[k][n];
  for (int x : xs) {
    ans *= c_fn(n, x);
  }
  cout << ans << endl;
}