Singers' Tour

2000ms

262144K

Description:

$$$n$$$ 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.

Input:

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 $$$2 \cdot 10^5$$$.

Output:

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.

Sample Input:

4
3
12 16 14
1
1
3
1 2 3
6
81 75 75 93 93 87

Sample Output:

YES
3 1 3 
YES
1 
NO
YES
5 5 4 1 4 5

Note:

Let's consider the $$$1$$$-st test case of the example:

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;
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;
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.

Informação

Provedor Codeforces

Código CF1618E