#include <bits/stdc++.h>
using namespace std;
long long MOD = 1000000007;
vector<int> pow_3(16, 1);
static inline int weigth(int n) {
  if (n == 0) return 0;
  int res = 0;
  while (n) {
    int q = __builtin_ffs(n) - 1;
    n ^= (1 << q);
    res += pow_3[q];
  }
  return res;
}
int main(void) {
  ios::sync_with_stdio(0);
  cin.tie(0);
  cout << fixed << setprecision(14);
  vector<long long> inv(2000003, 1);
  for (int i = 2; i < 2000003; i++) {
    inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;
  }
  for (int i = 1; i < 16; i++) pow_3[i] = 3 * pow_3[i - 1];
  int n;
  cin >> n;
  int N = 1 << n;
  vector<long long> a(n);
  for (int i = 0; i < n; i++) cin >> a[i];
  vector<long long> masks(n, 1);
  for (int i = 1; i < n; i++) masks[i] = 1 << i;
  vector<vector<long long>> defeat_prob(n, vector<long long>(N, 1));
  for (int i = 0; i < n; i++) {
    for (int j = 1; j < N; j++) {
      if (!(j & masks[i])) {
        int least_sign_j = __builtin_ffs(j) - 1;
        defeat_prob[i][j] =
            (defeat_prob[i][j & (j - 1)] * a[least_sign_j]) % MOD;
        defeat_prob[i][j] =
            (defeat_prob[i][j] * inv[a[least_sign_j] + a[i]]) % MOD;
      }
    }
  }
  int length = 1;
  for (int i = 0; i < n; i++) length *= 3;
  vector<long long> probs(length, 1);
  vector<long long> win_prob(N, 0);
  for (int m = 0; m < N; m++) {
    int w_m = weigth(m);
    for (int s = m; s; s = (s - 1) & m) {
      int x = m ^ s;
      int w_x = weigth(x);
      int w_s = weigth(s);
      if (s == m) {
        probs[w_s] = 0;
        continue;
      }
      int ss = s;
      long long prob = 1;
      while (ss) {
        int q = __builtin_ffs(ss) - 1;
        prob = (prob * defeat_prob[q][x]) % MOD;
        ss ^= masks[q];
      }
      probs[2 * w_x + w_s] = (probs[2 * w_x] * prob) % MOD;
      probs[2 * w_m] = (probs[2 * w_m] - probs[2 * w_x + w_s] + MOD) % MOD;
    }
  }
  long long expected = 0;
  for (int m = 0; m < N; m++) {
    int count = 0;
    int mm = m;
    while (mm) {
      int q = __builtin_ffs(mm) - 1;
      count++;
      mm ^= masks[q];
    }
    int s = weigth((N - 1) ^ m);
    expected = (expected + count * probs[2 * weigth(m) + s]) % MOD;
  }
  cout << expected << "\n";
}

### Prompt

Please formulate 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;
long long MOD = 1000000007;
vector<int> pow_3(16, 1);
static inline int weigth(int n) {
  if (n == 0) return 0;
  int res = 0;
  while (n) {
    int q = __builtin_ffs(n) - 1;
    n ^= (1 << q);
    res += pow_3[q];
  }
  return res;
}
int main(void) {
  ios::sync_with_stdio(0);
  cin.tie(0);
  cout << fixed << setprecision(14);
  vector<long long> inv(2000003, 1);
  for (int i = 2; i < 2000003; i++) {
    inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;
  }
  for (int i = 1; i < 16; i++) pow_3[i] = 3 * pow_3[i - 1];
  int n;
  cin >> n;
  int N = 1 << n;
  vector<long long> a(n);
  for (int i = 0; i < n; i++) cin >> a[i];
  vector<long long> masks(n, 1);
  for (int i = 1; i < n; i++) masks[i] = 1 << i;
  vector<vector<long long>> defeat_prob(n, vector<long long>(N, 1));
  for (int i = 0; i < n; i++) {
    for (int j = 1; j < N; j++) {
      if (!(j & masks[i])) {
        int least_sign_j = __builtin_ffs(j) - 1;
        defeat_prob[i][j] =
            (defeat_prob[i][j & (j - 1)] * a[least_sign_j]) % MOD;
        defeat_prob[i][j] =
            (defeat_prob[i][j] * inv[a[least_sign_j] + a[i]]) % MOD;
      }
    }
  }
  int length = 1;
  for (int i = 0; i < n; i++) length *= 3;
  vector<long long> probs(length, 1);
  vector<long long> win_prob(N, 0);
  for (int m = 0; m < N; m++) {
    int w_m = weigth(m);
    for (int s = m; s; s = (s - 1) & m) {
      int x = m ^ s;
      int w_x = weigth(x);
      int w_s = weigth(s);
      if (s == m) {
        probs[w_s] = 0;
        continue;
      }
      int ss = s;
      long long prob = 1;
      while (ss) {
        int q = __builtin_ffs(ss) - 1;
        prob = (prob * defeat_prob[q][x]) % MOD;
        ss ^= masks[q];
      }
      probs[2 * w_x + w_s] = (probs[2 * w_x] * prob) % MOD;
      probs[2 * w_m] = (probs[2 * w_m] - probs[2 * w_x + w_s] + MOD) % MOD;
    }
  }
  long long expected = 0;
  for (int m = 0; m < N; m++) {
    int count = 0;
    int mm = m;
    while (mm) {
      int q = __builtin_ffs(mm) - 1;
      count++;
      mm ^= masks[q];
    }
    int s = weigth((N - 1) ^ m);
    expected = (expected + count * probs[2 * weigth(m) + s]) % MOD;
  }
  cout << expected << "\n";
}

```