CF1478C 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 $ .

N/A

N/A

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 $ .