#include <bits/stdc++.h>
using namespace std;
constexpr int MAXN = 14;
int arr[MAXN];
int masks[1 << MAXN];
int add_mod(int a, int b) { return (a + b) % 1000000007; }
int mult_mod(int a, int b) { return (int)((((long long)a) * b) % 1000000007); }
int neg_mod(int a) { return (1000000007 - a) % 1000000007; }
long long powmod(int base, int e) {
  if (e == 0) return 1;
  long long tmp = powmod(base, e / 2);
  tmp *= tmp;
  tmp %= 1000000007;
  if (e & 1) {
    tmp *= base;
    tmp %= 1000000007;
  }
  return tmp;
}
int inv(int a) { return (int)powmod(a, 1000000007 - 2); }
int divi(int a, int b) { return mult_mod(a, inv(b)); }
int prob[MAXN][MAXN];
int noneness_cache[MAXN][1 << MAXN];
int32_t main() {
  ios_base::sync_with_stdio(false);
  int n;
  cin >> n;
  for (int i = 0; i < n; ++i) {
    cin >> arr[i];
  }
  for (int i = 0; i < n; ++i) {
    for (int j = 0; j < n; ++j) {
      prob[i][j] = divi(arr[i], add_mod(arr[i], arr[j]));
    }
  }
  for (int i = 0; i < n; ++i) {
    for (unsigned bm = 0; bm < (1u << n); ++bm) {
      int noneness = 1;
      for (int k = 0; k < n; ++k) {
        if ((1u << k) & bm) {
          noneness = mult_mod(noneness, prob[i][k]);
        }
      }
      noneness_cache[i][bm] = add_mod(1, neg_mod(noneness));
    }
  }
  int ans = 0;
  for (int i = 0; i < n; ++i) {
    for (unsigned bm = 0; bm != (1u << n); ++bm) {
      if ((bm & (1u << i)) == 0) {
        masks[bm] = 0;
      } else if (bm == (1u << i)) {
        masks[bm] = 1;
      } else {
        int total = 0;
        auto bm_popcnt = __builtin_popcount(bm);
        for (unsigned b2 = (bm - 1) & bm; b2 != 0; b2 = (b2 - 1) & bm) {
          int eff = masks[b2];
          unsigned diff = bm ^ b2;
          for (int j = 0; j < n; ++j) {
            if ((1 << j) & diff) {
              eff = mult_mod(eff, noneness_cache[j][b2]);
            }
          }
          auto b2_popcnt = __builtin_popcount(b2);
          if ((bm_popcnt - b2_popcnt) & 1) {
            total = add_mod(total, eff);
          } else {
            total = add_mod(total, neg_mod(eff));
          }
        }
        masks[bm] = total;
      }
    }
    ans = add_mod(ans, masks[(1u << n) - 1u]);
    for (unsigned bm = 0; bm != (1u << n); ++bm) {
      masks[bm] = 0;
    }
  }
  cout << ans << '\n';
}

### Prompt

Generate a Cpp solution to the following problem:

<image>

William is not only interested in trading but also in betting on sports matches. n teams participate in each match. Each team is characterized by strength a_i. Each two teams i < j play with each other exactly once. Team i wins with probability (a_i)/(a_i + a_j) and team j wins with probability (a_j)/(a_i + a_j).

The team is called a winner if it directly or indirectly defeated all other teams. Team a defeated (directly or indirectly) team b if there is a sequence of teams c_1, c_2, ... c_k such that c_1 = a, c_k = b and team c_i defeated team c_{i + 1} for all i from 1 to k - 1. Note that it is possible that team a defeated team b and in the same time team b defeated team a.

William wants you to find the expected value of the number of winners.

Input

The first line contains a single integer n (1 ≤ n ≤ 14), which is the total number of teams participating in a match.

The second line contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^6) — the strengths of teams participating in a match.

Output

Output a single integer — the expected value of the number of winners of the tournament modulo 10^9 + 7.

Formally, let M = 10^9+7. It can be demonstrated that the answer can be presented as a irreducible fraction p/q, where p and q are integers and q not ≡ 0 \pmod{M}. Output a single integer equal to p ⋅ q^{-1} mod M. In other words, output an integer x such that 0 ≤ x < M and x ⋅ q ≡ p \pmod{M}.

Examples

Input


2
1 2


Output


1


Input


5
1 5 2 11 14


Output


642377629

Note

To better understand in which situation several winners are possible let's examine the second test:

One possible result of the tournament is as follows (a → b means that a defeated b):

  * 1 → 2 
  * 2 → 3 
  * 3 → 1 
  * 1 → 4 
  * 1 → 5 
  * 2 → 4 
  * 2 → 5 
  * 3 → 4 
  * 3 → 5 
  * 4 → 5 



Or more clearly in the picture:

<image>

In this case every team from the set \{ 1, 2, 3 \} directly or indirectly defeated everyone. I.e.:

  * 1st defeated everyone because they can get to everyone else in the following way 1 → 2, 1 → 2 → 3, 1 → 4, 1 → 5. 
  * 2nd defeated everyone because they can get to everyone else in the following way 2 → 3, 2 → 3 → 1, 2 → 4, 2 → 5. 
  * 3rd defeated everyone because they can get to everyone else in the following way 3 → 1, 3 → 1 → 2, 3 → 4, 3 → 5. 



Therefore the total number of winners is 3.

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
constexpr int MAXN = 14;
int arr[MAXN];
int masks[1 << MAXN];
int add_mod(int a, int b) { return (a + b) % 1000000007; }
int mult_mod(int a, int b) { return (int)((((long long)a) * b) % 1000000007); }
int neg_mod(int a) { return (1000000007 - a) % 1000000007; }
long long powmod(int base, int e) {
  if (e == 0) return 1;
  long long tmp = powmod(base, e / 2);
  tmp *= tmp;
  tmp %= 1000000007;
  if (e & 1) {
    tmp *= base;
    tmp %= 1000000007;
  }
  return tmp;
}
int inv(int a) { return (int)powmod(a, 1000000007 - 2); }
int divi(int a, int b) { return mult_mod(a, inv(b)); }
int prob[MAXN][MAXN];
int noneness_cache[MAXN][1 << MAXN];
int32_t main() {
  ios_base::sync_with_stdio(false);
  int n;
  cin >> n;
  for (int i = 0; i < n; ++i) {
    cin >> arr[i];
  }
  for (int i = 0; i < n; ++i) {
    for (int j = 0; j < n; ++j) {
      prob[i][j] = divi(arr[i], add_mod(arr[i], arr[j]));
    }
  }
  for (int i = 0; i < n; ++i) {
    for (unsigned bm = 0; bm < (1u << n); ++bm) {
      int noneness = 1;
      for (int k = 0; k < n; ++k) {
        if ((1u << k) & bm) {
          noneness = mult_mod(noneness, prob[i][k]);
        }
      }
      noneness_cache[i][bm] = add_mod(1, neg_mod(noneness));
    }
  }
  int ans = 0;
  for (int i = 0; i < n; ++i) {
    for (unsigned bm = 0; bm != (1u << n); ++bm) {
      if ((bm & (1u << i)) == 0) {
        masks[bm] = 0;
      } else if (bm == (1u << i)) {
        masks[bm] = 1;
      } else {
        int total = 0;
        auto bm_popcnt = __builtin_popcount(bm);
        for (unsigned b2 = (bm - 1) & bm; b2 != 0; b2 = (b2 - 1) & bm) {
          int eff = masks[b2];
          unsigned diff = bm ^ b2;
          for (int j = 0; j < n; ++j) {
            if ((1 << j) & diff) {
              eff = mult_mod(eff, noneness_cache[j][b2]);
            }
          }
          auto b2_popcnt = __builtin_popcount(b2);
          if ((bm_popcnt - b2_popcnt) & 1) {
            total = add_mod(total, eff);
          } else {
            total = add_mod(total, neg_mod(eff));
          }
        }
        masks[bm] = total;
      }
    }
    ans = add_mod(ans, masks[(1u << n) - 1u]);
    for (unsigned bm = 0; bm != (1u << n); ++bm) {
      masks[bm] = 0;
    }
  }
  cout << ans << '\n';
}

```