CF1822D 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.