CF1779B.MKnez's ConstructiveForces Task

MKnez wants to construct an array $s_1,s_2, \ldots , s_n$ satisfying the following conditions:

• Each element is an integer number different from $0$ ;
• For each pair of adjacent elements their sum is equal to the sum of the whole array.

More formally, $s_i \neq 0$ must hold for each $1 \leq i \leq n$ . Moreover, it must hold that $s_1 + s_2 + \cdots + s_n = s_i + s_{i+1}$ for each $1 \leq i < n$ .

Help MKnez to construct an array with these properties or determine that it does not exist.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \leq t \leq 100$ ). The description of the test cases follows.

The only line of each test case contains a single integer $n$ ( $2 \leq n \leq 1000$ ) — the length of the array.

For each test case, print "YES" if an array of length $n$ satisfying the conditions exists. Otherwise, print "NO". If the answer is "YES", on the next line print a sequence $s_1,s_2, \ldots, s_n$ satisfying the conditions. Each element should be a non-zero integer in the range $[-5000,5000]$ , i. e. $-5000 \leq s_i \leq 5000$ and $s_i \neq 0$ should hold for each $1 \leq i \leq n$ .

It can be proved that if a solution exists then there also exists one which satisfies the additional constraints on the range.

If there are several correct answers, print any of them.

In the first test case, $[9,5]$ is a valid answer since $9+5$ (the sum of the two adjacent elements $s_1+s_2$ ) is equal to $9+5$ (the sum of all elements). Other solutions include $[6,-9], [-1,-2], [-5000,5000], \ldots$

For the second test case, let us show why some arrays do not satisfy the constraints:

• $[1,1,1]$ — $s_1+s_2 = 1+1 = 2$ and $s_1+s_2+s_3=1+1+1 = 3$ differ;
• $[1,-1,1]$ — $s_1+s_2=1+(-1)=0$ and $s_1+s_2+s_3=1+(-1)+1 = 1$ differ;
• $[0,0,0]$ — The array $s$ cannot contain a $0$ .

This is not a proof, but it can be shown that the answer is "NO".