Students from Antarctic Penguin Language School are studying sequence.

We call a sequence  of length  beautiful when all the following conditions are true:

•  
•  For each (), .
•  For each , , where is a given sequence, and  represents bitwise XOR.

You need to find the -th smallest lexicographically legal beautiful sequence .

输入描述:

Each test contains multiple test cases. The first line contains the number of test cases . The description of the test cases follows.

The first line contains two integers ,  (, ).

The second line contains  integers  ().

It is guaranteed that the sum of  over all test cases does not exceed .

输出描述:

For each test case, output your operations in the following format.

If the number of beautiful sequences is less than , output one line containing an integer .

Otherwise, output one line containing  integers  separated by spaces --- the -th smallest lexicographically legal beautiful sequence.

3
3 3
1 2
5 1
1 2 1 2
2 1
0

8 9 11
-1
0 0