Nezzar and Symmetric Array

题意翻译

定义长度为 $2 \times n$ 的数组 $a$ ，满足： $a$ 中的元素不重复。 并且对于任意一个下标$i(1 \leq i \leq 2 \cdot n, i \ne j)$，都能找到一个下标 $j$，使得 $a_i = -a_j$ 。 现在给定一个数组 $d$，其中 $$d_i = \sum_{j=1}^{2n}|a_i -a_j|$$ 问能不能通过数组 $d$ 构造出数组 $a$ 。 能构造出输出 `YES`，不能就输出 `NO` 有 $T$ 组数据，其中 $1\le T \le10^5,1\le n \le10^5,1\le d_i\le10^{12}$ 并且保证 $(\sum_{n\in\ \text one\ \text text\ \text case} n ) \le\ 10^5$

题目描述

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 $ .

输入输出格式

输入格式

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 $ .

输出格式

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).

输入输出样例

输入样例 #1

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

输出样例 #1

YES
NO
NO
NO
NO
YES

说明

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 $ .