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문제

The boboness of a sequence of integers $(x_1, x_2, \dots, x_n)$ is $\prod\limits_{i = 3}^n w(\max\{x_{i - 2}, x_{i - 1}, x_i\})$. Here, $1 \leq x_i \leq n$, and the values $w(1), w(2), \dots, w(n)$ are given.

Bobo would like to know the sum of boboness of all sequences satisfying $1 \leq x_i \leq n$. As this sum can be very large, he is interested only in the answer modulo $(10^9+7)$.

입력

The input contains zero or more test cases, and is terminated by end-of-file. For each test case:

The first line contains an integer $n$ ($3 \leq n \leq 2000$).

The second line contains $n$ integers $w(1), w(2), \dots, w(n)$ ($1 \leq w(i) \leq 10^9$).

It is guaranteed that the sum of $n$ does not exceed $2000$.

출력

For each test case, output an integer which denotes the sum taken modulo $(10^9+7)$.

예제 입력 1

3
1 2 3
4
1 1 1 1

예제 출력 1

72
256