Problem Statement

Given is an integer sequence of length $N$: $A_1, A_2, \cdots, A_N$.

An integer sequence $X$, which is also of length $N$, will be chosen randomly by independently choosing $X_i$ from a uniform distribution on the integers $1, 2, \ldots, A_i$ for each $i (1 \leq i \leq N)$.

Compute the expected value of the length of the longest increasing subsequence of this sequence $X$, modulo $1000000007$.

More formally, under the constraints of the problem, we can prove that the expected value can be represented as a rational number, that is, an irreducible fraction $\frac{P}{Q}$, and there uniquely exists an integer $R$ such that $R \times Q \equiv P \pmod {1000000007}$ and $0 \leq R < 1000000007$, so print this integer $R$.

Note

A subsequence of a sequence $X$ is a sequence obtained by extracting some of the elements of $X$ and arrange them without changing the order. The longest increasing subsequence of a sequence $X$ is the sequence of the greatest length among the strictly increasing subsequences of $X$.

Constraints

• $1 \leq N \leq 6$
• $1 \leq A_i \leq 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 \cdots A_N

Output

Print the expected value modulo $1000000007$.