A15814. Sequence Master

For some positive integer $m$ , YunQian considers an array $q$ of $2m$ (possibly negative) integers good, if and only if for every possible subsequence of $q$ that has length $m$ , the product of the $m$ elements in the subsequence is equal to the sum of the $m$ elements that are not in the subsequence. Formally, let $U=\{1,2,\ldots,2m\}$ . For all sets $S \subseteq U$ such that $|S|=m$ , $\prod\limits_{i \in S} q_i = \sum\limits_{i \in U \setminus S} q_i$ .

Define the distance between two arrays $a$ and $b$ both of length $k$ to be $\sum\limits_{i=1}^k|a_i-b_i|$ .

You are given a positive integer $n$ and an array $p$ of $2n$ integers.

Find the minimum distance between $p$ and $q$ over all good arrays $q$ of length $2n$ . It can be shown for all positive integers $n$ , at least one good array exists. Note that you are not required to construct the array $q$ that achieves this minimum distance.

The first line contains a single integer $t$ ( $1\le t\le 10^4$ ) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $n$ ( $1\le n\le 2\cdot10^5$ ).

The second line of each test case contains $2n$ integers $p_1, p_2, \ldots, p_{2n}$ ( $|p_i| \le 10^9$ ).

It is guaranteed that the sum of $n$ over all test cases does not exceed $2\cdot 10^5$ .

For each test case, output the minimum distance between $p$ and a good $q$ .

In the first test case, it is optimal to let $q=[6,6]$ .

In the second test case, it is optimal to let $q=[2,2,2,2]$ .