In the End

The board is infinite in one direction. It has $n$ rows and an infinite number of columns, with a first (leftmost) column but no last column. Each column contains exactly one cell with a cake. The cake is in row $i$ with probability $p_i$, and all $p_i$ sum to $1$. Cake positions in different columns are independent. A robot starts in some cell of the first column. If the robot is at cell $(x, y)$ (row $x$, column $y$), in one step it can move to $(x, y + 1)$, $(x - 1, y + 1)$, or $(x + 1, y + 1)$, whenever that cell exists. When the robot visits a cell with a cake, it collects that cake. The order of events is as follows: first all cakes are placed according to the given probabilities, then the configuration is given to the robot, and then the robot chooses a starting cell and a movement plan to maximize the number of cakes it collects. For each board of size $n \times m$ with $m = 1, 2, \ldots$, let $f(m)$ be the expected average number of cakes collected per step under an optimal route. Compute $\lim_{m \rightarrow \infty} \frac{f(m)}{m}$.

The first line contains an integer $n$ ($1 \le n \le 6$), the number of rows. The next line contains $n$ real numbers $p_i$ ($0 \le p_i \le 1$) with at most one digit after the decimal point. Their sum is $1$.

The answer can be written as a fraction $\frac{P}{Q}$ with positive integers $P$ and $Q$. Print $(P \cdot Q^{-1}) \bmod (10^9 + 7)$.