CF1618E Singers' Tour

$ n $ towns are arranged in a circle sequentially. The towns are numbered from $ 1 $ to $ n $ in clockwise order. In the $ i $ -th town, there lives a singer with a repertoire of $ a_i $ minutes for each $ i \in [1, n] $ . Each singer visited all $ n $ towns in clockwise order, starting with the town he lives in, and gave exactly one concert in each town. In addition, in each town, the $ i $ -th singer got inspired and came up with a song that lasts $ a_i $ minutes. The song was added to his repertoire so that he could perform it in the rest of the cities. Hence, for the $ i $ -th singer, the concert in the $ i $ -th town will last $ a_i $ minutes, in the $ (i + 1) $ -th town the concert will last $ 2 \cdot a_i $ minutes, ..., in the $ ((i + k) \bmod n + 1) $ -th town the duration of the concert will be $ (k + 2) \cdot a_i $ , ..., in the town $ ((i + n - 2) \bmod n + 1) $ — $ n \cdot a_i $ minutes. You are given an array of $ b $ integer numbers, where $ b_i $ is the total duration of concerts in the $ i $ -th town. Reconstruct any correct sequence of positive integers $ a $ or say that it is impossible.

The first line contains one integer $ t $ $ (1 \le t \le 10^3 $ ) — the number of test cases. Then the test cases follow. Each test case consists of two lines. The first line contains a single integer $ n $ ( $ 1 \le n \le 4 \cdot 10^4 $ ) — the number of cities. The second line contains $ n $ integers $ b_1, b_2, \dots, b_n $ ( $ 1 \le b_i \le 10^{9} $ ) — the total duration of concerts in $ i $ -th city. The sum of $ n $ over all test cases does not exceed $ 2 \cdot 10^5 $ .

For each test case, print the answer as follows: If there is no suitable sequence $ a $ , print NO. Otherwise, on the first line print YES, on the next line print the sequence $ a_1, a_2, \dots, a_n $ of $ n $ integers, where $ a_i $ ( $ 1 \le a_i \le 10^{9} $ ) is the initial duration of repertoire of the $ i $ -th singer. If there are multiple answers, print any of them.

Let's consider the $ 1 $ -st test case of the example: 1. the $ 1 $ -st singer in the $ 1 $ -st city will give a concert for $ 3 $ minutes, in the $ 2 $ -nd — for $ 6 $ minutes, in the $ 3 $ -rd — for $ 9 $ minutes; 2. the $ 2 $ -nd singer in the $ 1 $ -st city will give a concert for $ 3 $ minutes, in the $ 2 $ -nd — for $ 1 $ minute, in the $ 3 $ -rd - for $ 2 $ minutes; 3. the $ 3 $ -rd singer in the $ 1 $ -st city will give a concert for $ 6 $ minutes, in the $ 2 $ -nd — for $ 9 $ minutes, in the $ 3 $ -rd — for $ 3 $ minutes.