Super-Permutation

2000ms

262144K

Description:

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 $$$-1$$$.

Input:

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

Output:

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.

Sample Input:

Sample Output:

1
2 1
-1
6 5 2 3 4 1

Informação

Provedor Codeforces

Código CF1822D