B"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. The first line contains a single integer n ( 1 <= q n <= q 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 <= q a_i <= q 10^6 ) -- the strengths of teams participating in a match. 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 <= x < M and x cdot q equiv p pmod{M} . 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 ): Or more clearly in the picture: In this case every team from the set { 1, 2, 3 } di"...