Half-sum
Description
You're given a multiset of non-negative integers $\{a_1, a_2, \dots, a_n\}$.
In one step you take two elements $x$ and $y$ of the multiset, remove them and insert their mean value $\frac{x + y}{2}$ back into the multiset.
You repeat the step described above until you are left with only two numbers $A$ and $B$. What is the maximum possible value of their absolute difference $|A-B|$?
Since the answer is not an integer number, output it modulo $10^9+7$.
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 $10^6$.
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 $x \cdot q \equiv p \pmod{M}$.
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 $x \cdot q \equiv p \pmod{M}$.
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
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:
- Substitute $1$ and $2$ with $1.5$;
- Substitute $10$ and $11$ with $10.5$;
- The difference between $1.5$ and $10.5$ is $9$.
In the third case, the exact answer is $\frac{3}{2}$, and $500\,000\,005 \cdot 2 \equiv 3 \pmod{10^9+7}$.