A14166. Nezzar and Symmetric Array

Long time ago there was a symmetric array $a_1,a_2,\ldots,a_{2n}$ consisting of $2n$ distinct integers. Array $a_1,a_2,\ldots,a_{2n}$ is called symmetric if for each integer $1 \le i \le 2n$ , there exists an integer $1 \le j \le 2n$ such that $a_i = -a_j$ .

For each integer $1 \le i \le 2n$ , Nezzar wrote down an integer $d_i$ equal to the sum of absolute differences from $a_i$ to all integers in $a$ , i. e. $d_i = \sum_{j = 1}^{2n} {|a_i - a_j|}$ .

Now a million years has passed and Nezzar can barely remember the array $d$ and totally forget $a$ . Nezzar wonders if there exists any symmetric array $a$ consisting of $2n$ distinct integers that generates the array $d$ .

The first line contains a single integer $t$ ( $1 \le t \le 10^5$ ) — the number of test cases.

The first line of each test case contains a single integer $n$ ( $1 \le n \le 10^5$ ).

The second line of each test case contains $2n$ integers $d_1, d_2, \ldots, d_{2n}$ ( $0 \le d_i \le 10^{12}$ ).

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

For each test case, print "YES" in a single line if there exists a possible array $a$ . Otherwise, print "NO".

You can print letters in any case (upper or lower).

In the first test case, $a=[1,-3,-1,3]$ is one possible symmetric array that generates the array $d=[8,12,8,12]$ .

In the second test case, it can be shown that there is no symmetric array consisting of distinct integers that can generate array $d$ .