C2. XOR-convenience (Hard Version)
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

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.

Input

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

Output

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.

Example
Input
2
3
4
Output
2 1 3
-1
Note

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.