CF2189C2 XOR-convenience (Hard Version)

This is the hard version of the problem. The difference between the versions is that in this version, $ 1 \le i \le n-1 $ . Note that a correct solution for the hard version is not necessarily a correct solution for the easy version. Given a positive integer $ n $ . Find some permutation $ ^{\text{∗}} $ $ p $ of length $ n $ such that for every $ i $ ( $ \style{color:red}{1 \le i \le n-1} $ ) there exists $ j $ ( $ \style{color:red}{i \le j \le n} $ ) such that $ p_i = p_j \oplus i $ $ ^{\text{†}} $ , or determine that it does not exist. $ ^{\text{∗}} $ A permutation of length $ n $ is an array consisting of $ n $ distinct integers from $ 1 $ to $ n $ in arbitrary order. For example, $ [2,3,1,5,4] $ is a permutation, but $ [1,2,2] $ is not a permutation ( $ 2 $ appears twice in the array), and $ [1,3,4] $ is also not a permutation ( $ n=3 $ but there is $ 4 $ in the array). $ ^{\text{†}} $ $ \oplus $ denotes the [bitwise XOR operation](https://en.wikipedia.org/wiki/Bitwise_operation#XOR).

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 10^4 $ ). The description of the test cases follows. The only line of each test case contains one positive integer $ n $ ( $ 3 \leq n \leq 2 \cdot 10^5 $ ) — the length of the permutation. It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 2\cdot 10^5 $ .

For each test case, if there is a suitable permutation, output $ n $ integers $ p_1,p_2,\ldots,p_n $ — the permutation $ p $ . Otherwise, output $ -1 $ . If multiple solutions exist, output any of them.

In the first test case, the permutation $ p = [2,1,3] $ fulfills the condition, as $ p_2 = 1 $ and $ p_3 \oplus 2 = 1 $ . In the second test case, there is no suitable permutation.