#include <bits/stdc++.h>
using namespace std;
const int mod = 1e9 + 7;
long long fpow(long long a, long long b) {
  long long r = 1;
  for (a %= mod; b; b >>= 1) {
    if (b & 1) r = r * a % mod;
    a = a * a % mod;
  }
  return r;
}
int main() {
  ios::sync_with_stdio(0);
  cin.tie(0);
  cout.tie(0);
  int n;
  cin >> n;
  vector<long long> a(n);
  for (int i = 0; i < n; ++i) cin >> a[i];
  vector<vector<long long>> win(n, vector<long long>(n));
  for (int i = 0; i < n; ++i) {
    for (int j = 0; j < n; ++j) {
      if (i == j) continue;
      win[i][j] = a[i] * fpow(a[i] + a[j], mod - 2) % mod;
    }
  }
  vector<long long> dp(1 << n);
  for (int i = 1; i < 1 << n; ++i) {
    dp[i] = 1;
    for (int j = (i - 1) & i; j; j = (j - 1) & i) {
      long long tmp = 1;
      for (int k = 0; k < n; ++k) {
        if ((i >> k & 1) && (j >> k & 1))
          for (int p = 0; p < n; ++p) {
            if ((i >> p & 1) && !(j >> p & 1)) {
              tmp = tmp * win[k][p] % mod;
            }
          }
      }
      dp[i] -= dp[j] * tmp % mod;
    }
    dp[i] %= mod;
  }
  long long ans = 0;
  for (int i = 1; i < 1 << n; ++i) {
    long long tmp = 1;
    for (int j = 0; j < n; ++j) {
      if (i >> j & 1)
        for (int k = 0; k < n; ++k) {
          if (!(i >> k & 1)) tmp = tmp * win[j][k] % mod;
        }
    }
    ans += tmp * dp[i] % mod * __builtin_popcount(i) % mod;
  }
  ans %= mod;
  if (ans < 0) ans += mod;
  cout << ans << '\n';
}

### Prompt

Your challenge is to write 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;
const int mod = 1e9 + 7;
long long fpow(long long a, long long b) {
  long long r = 1;
  for (a %= mod; b; b >>= 1) {
    if (b & 1) r = r * a % mod;
    a = a * a % mod;
  }
  return r;
}
int main() {
  ios::sync_with_stdio(0);
  cin.tie(0);
  cout.tie(0);
  int n;
  cin >> n;
  vector<long long> a(n);
  for (int i = 0; i < n; ++i) cin >> a[i];
  vector<vector<long long>> win(n, vector<long long>(n));
  for (int i = 0; i < n; ++i) {
    for (int j = 0; j < n; ++j) {
      if (i == j) continue;
      win[i][j] = a[i] * fpow(a[i] + a[j], mod - 2) % mod;
    }
  }
  vector<long long> dp(1 << n);
  for (int i = 1; i < 1 << n; ++i) {
    dp[i] = 1;
    for (int j = (i - 1) & i; j; j = (j - 1) & i) {
      long long tmp = 1;
      for (int k = 0; k < n; ++k) {
        if ((i >> k & 1) && (j >> k & 1))
          for (int p = 0; p < n; ++p) {
            if ((i >> p & 1) && !(j >> p & 1)) {
              tmp = tmp * win[k][p] % mod;
            }
          }
      }
      dp[i] -= dp[j] * tmp % mod;
    }
    dp[i] %= mod;
  }
  long long ans = 0;
  for (int i = 1; i < 1 << n; ++i) {
    long long tmp = 1;
    for (int j = 0; j < n; ++j) {
      if (i >> j & 1)
        for (int k = 0; k < n; ++k) {
          if (!(i >> k & 1)) tmp = tmp * win[j][k] % mod;
        }
    }
    ans += tmp * dp[i] % mod * __builtin_popcount(i) % mod;
  }
  ans %= mod;
  if (ans < 0) ans += mod;
  cout << ans << '\n';
}

```