Integration

题号：NC51063
时间限制：C/C++/Rust/Pascal 2秒，其他语言4秒
空间限制：C/C++/Rust/Pascal 512 M，其他语言1024 M
64bit IO Format: %lld

题目描述

Bobo knows that $\int_{0}^{\infty} \frac{1}{1 + x^2}\ \mathrm{d}x = \frac{\pi}{2}.$

Given n distinct positive integers $a_1, a_2, \dots, a_n$ , find the value of
$\frac{1}{\pi} \int_{0}^{\infty} \frac{1}{\prod_{i = 1}^n(a_i^2 + x^2)}\ \mathrm{d}x.$

It can be proved that the value is a rational number $\frac{p}{q}$ .
Print the result as $(p \cdot q^{-1}) \bmod (10^9+7)$ .

输入描述:

The input consists of several test cases and is terminated by end-of-file.

The first line of each test case contains an integer n.
The second line contains n integers .

* 
* 
*  are distinct.
* The sum of  does not exceed .

输出描述:

For each test case, print an integer which denotes the result.

示例1

输入

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输出

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500000004
250000002
83333334