F. Sports Betting
time limit per test
4 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

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 $\frac{a_i}{a_i + a_j}$ and team $j$ wins with probability $\frac{a_j}{a_i + a_j}$.

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 $n$ ($1 \leq n \leq 14$), which is the total number of teams participating in a match.

The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \leq a_i \leq 10^6$)  — 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 $\frac{p}{q}$, where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{M}$. Output a single integer equal to $p \cdot q^{-1} \bmod M$. In other words, output an integer $x$ such that $0 \le x < M$ and $x \cdot q \equiv p \pmod{M}$.

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 ($a \rightarrow b$ means that $a$ defeated $b$):

• $1 \rightarrow 2$
• $2 \rightarrow 3$
• $3 \rightarrow 1$
• $1 \rightarrow 4$
• $1 \rightarrow 5$
• $2 \rightarrow 4$
• $2 \rightarrow 5$
• $3 \rightarrow 4$
• $3 \rightarrow 5$
• $4 \rightarrow 5$

Or more clearly in the picture:

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

• $1$st defeated everyone because they can get to everyone else in the following way $1 \rightarrow 2$, $1 \rightarrow 2 \rightarrow 3$, $1 \rightarrow 4$, $1 \rightarrow 5$.
• $2$nd defeated everyone because they can get to everyone else in the following way $2 \rightarrow 3$, $2 \rightarrow 3 \rightarrow 1$, $2 \rightarrow 4$, $2 \rightarrow 5$.
• $3$rd defeated everyone because they can get to everyone else in the following way $3 \rightarrow 1$, $3 \rightarrow 1 \rightarrow 2$, $3 \rightarrow 4$, $3 \rightarrow 5$.

Therefore the total number of winners is $3$.