Nezzar and Symmetric Array

2000ms

524288K

Description:

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 $$$d$$$.

Input:

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 $$$10^5$$$.

Output:

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).

Sample Input:

6
2
8 12 8 12
2
7 7 9 11
2
7 11 7 11
1
1 1
4
40 56 48 40 80 56 80 48
6
240 154 210 162 174 154 186 240 174 186 162 210

Sample Output:

YES
NO
NO
NO
NO
YES

Note:

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 $$$d$$$.

Informação

Provedor Codeforces

Código CF1478C