CF1817E.Half-sum

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 $x \cdot q \equiv p \pmod{M}$ .

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 $500\,000\,005 \cdot 2 \equiv 3 \pmod{10^9+7}$ .