A15932 | Super-Permutation

A permutation is a sequence $n$ integers, where each integer from $1$ to $n$ appears exactly once. For example, $[1]$ , $[3,5,2,1,4]$ , $[1,3,2]$ are permutations, while $[2,3,2]$ , $[4,3,1]$ , $[0]$ are not.

Given a permutation $a$ , we construct an array $b$ , where $b_i = (a_1 + a_2 +~\dots~+ a_i) \bmod n$ .

A permutation of numbers $[a_1, a_2, \dots, a_n]$ is called a super-permutation if $[b_1 + 1, b_2 + 1, \dots, b_n + 1]$ is also a permutation of length $n$ .

Grisha became interested whether a super-permutation of length $n$ exists. Help him solve this non-trivial problem. Output any super-permutation of length $n$ , if it exists. Otherwise, output $-1$ .

The first line contains a single integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases. The description of the test cases follows.

Each test case consists of a single line containing one integer $n$ ( $1 \le n \le 2 \cdot 10^5$ ) — the length of the desired permutation.

The sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ .

For each test case, output in a separate line:

- $n$ integers — a super-permutation of length $n$ , if it exists.
- $-1$ , otherwise.

If there are several suitable permutations, output any of them.