description: <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.
solution: #include <bits/stdc++.h>
  using namespace std;
  inline int read() {
    int res = 0, f = 1;
    char ch;
    while (ch = getchar(), ch < '0' || ch > '9')
      if (ch == '-') f = -1;
    while (ch >= '0' && ch <= '9') res = res * 10 + ch - '0', ch = getchar();
    return res * f;
  }
  const int N = 14, M = 4782969, P = 1e9 + 7;
  inline int qpow(int a, int b) {
    int ans = 1, base = a;
    while (b) {
      if (b & 1) ans = 1ll * ans * base % P;
      base = 1ll * base * base % P;
      b >>= 1;
    }
    return ans;
  }
  int rev[1 << N], cnt[1 << N], n, A, a[N];
  int F[1 << N], G[1 << N], v[N][1 << N];
  inline int get(int S, int T) {
    if (!S) return 1;
    if (G[S]) return G[S];
    return G[S] = 1ll * get(S & (S - 1), T) * v[rev[S & -S]][T] % P;
  }
  int main() {
    n = read(), A = (1 << n) - 1;
    for (int i = 0; i < n; ++i) a[i] = read(), rev[1 << i] = i;
    for (int i = 0; i < n; ++i) {
      v[i][0] = 1;
      for (int S = 1; S < (1 << n); ++S)
        v[i][S] = 1ll * v[i][S & (S - 1)] * a[i] % P *
                  qpow((a[rev[S & -S]] + a[i]) % P, P - 2) % P;
    }
    for (int S = 1; S < (1 << n); ++S) {
      int tmp = 0;
      cnt[S] = cnt[S >> 1] + (S & 1);
      for (int T = (S - 1) & S; T; T = (T - 1) & S)
        (tmp += 1ll * F[T] * get(S ^ T, A ^ S) % P) %= P;
      F[S] = (get(S, A ^ S) - tmp + P) % P;
      for (int T = S; T; T = (T - 1) & S) G[T] = 0;
    }
    int res = 0;
    for (int S = 1; S < (1 << n); ++S) (res += 1ll * F[S] * cnt[S] % P) %= P;
    printf("%d", res);
    return 0;
  }
  