Problem Statement

Given is an integer sequence of length $N$: A1​,A2​,⋯,AN​.

An integer sequence $X$, which is also of length $N$, will be chosen randomly by independently choosing Xi​ from a uniform distribution on the integers 1,2,…,Ai​ for each $i(1\le i\le N)$.

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 QP​, and there uniquely exists an integer $R$ such that $R\times Q\equiv P(\mathrm{m}\mathrm{o}\mathrm{d}1000000007)$ and $0\le R<1000000007$, so print this integer $R$.

Notes

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

• $1\le N\le 6$
• 1≤Ai​≤109
• All values in input are integers.

Sample Input 1

3
1 2 3

Sample Output 1

2

$X$ becomes one of the following, with probability $1/6$ for each:

• $X=(1,1,1)$, for which the length of the longest increasing subsequence is $1$;
• $X=(1,1,2)$, for which the length of the longest increasing subsequence is $2$;
• $X=(1,1,3)$, for which the length of the longest increasing subsequence is $2$;
• $X=(1,2,1)$, for which the length of the longest increasing subsequence is $2$;
• $X=(1,2,2)$, for which the length of the longest increasing subsequence is $2$;
• $X=(1,2,3)$, for which the length of the longest increasing subsequence is $3$.

Thus, the expected value of the length of the longest increasing subsequence is $(1+2+2+2+2+3)/6\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}1000000007)$.

Sample Input 2

3
2 1 2

Sample Output 2

500000005

$X$ becomes one of the following, with probability $1/4$ for each:

• $X=(1,1,1)$, for which the length of the longest increasing subsequence is $1$;
• $X=(1,1,2)$, for which the length of the longest increasing subsequence is $2$;
• $X=(2,1,1)$, for which the length of the longest increasing subsequence is $1$;
• $X=(2,1,2)$, for which the length of the longest increasing subsequence is $2$.

Thus, the expected value of the length of the longest increasing subsequence is $(1+2+1+2)/4=3/2$. Since $2\times 500000005\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}1000000007)$, we should print $500000005$.

Sample Input 3

6
8 6 5 10 9 14

Sample Output 3

959563502