Half-sum

Description:

Input:

Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 100$$$). Description of the test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \le n \le 10^6$$$) — the size of the multiset.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$0 \le a_i \le 10^9$$$) — the elements of the multiset.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^6$$$.

Output:

For each test case, output a single integer, the answer to the problem modulo $$$10^9+7$$$.

Formally, let $$$M = 10^9+7$$$. It can be shown that the answer can be expressed as an irreducible fraction $$$\frac{p}{q}$$$, where $$$p$$$ and $$$q$$$ are integers and $$$q \not \equiv 0 \pmod{M}$$$. Output the integer equal to $$$p \cdot q^{-1} \bmod M$$$. In other words, output an integer $$$x$$$ such that $$$0 \le x < M$$$ and $$$x \cdot q \equiv p \pmod{M}$$$.

Sample Input:

5
2
7 3
4
1 2 10 11
3
1 2 3
6
64 32 64 16 64 0
4
1 1 1 1

Sample Output:

4
9
500000005
59
0

Note:

In the first case, you can't do any operations, so the answer is $$$|7-3|=4$$$.

In the second case, one of the optimal sequence of operations:

1. Substitute $$$1$$$ and $$$2$$$ with $$$1.5$$$;
2. Substitute $$$10$$$ and $$$11$$$ with $$$10.5$$$;
3. The difference between $$$1.5$$$ and $$$10.5$$$ is $$$9$$$.

In the third case, the exact answer is $$$\frac{3}{2}$$$, and $$$500\,000\,005 \cdot 2 \equiv 3 \pmod{10^9+7}$$$.