F2. XOR Transformations (Hard Version)
time limit per test
4 seconds
memory limit per test
512 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, the constraints on $$$n$$$ and $$$q$$$ are higher. You can make hacks only if you have solved all versions of this problem.

For an array $$$b$$$ consisting of $$$m$$$ integers, define a transformation as follows:

  • Write down the values $$$b_i\oplus b_j$$$ for all $$$1\le i\lt j\le m$$$, where $$$\oplus$$$ denotes the bitwise XOR operation.
  • Take the $$$m$$$ smallest among the written values.
  • Replace the elements of the array with the taken $$$m$$$ values.

For example, consider the transformation of the array $$$[6, 7, 8, 9, 15]$$$. We write down the values $$$1, 1, 6, 7, 8, 9, 14, 14, 15, 15$$$. After the transformation, the array becomes $$$[1, 1, 6, 7, 8]$$$ — the $$$5$$$ smallest elements.

You are given an array $$$a$$$ consisting of $$$n$$$ non-negative integers. Let $$$\max(a)$$$ denote the maximum element of the array $$$a$$$, and $$$\min(a)$$$ — the minimum. Your task is to answer $$$q$$$ queries, each of which gives you one integer $$$x$$$. For each query, find the value of $$$\max(a) - \min(a)$$$ after $$$x$$$ transformations on the array. Note that the queries are independent, i.e. before each query, the array $$$a$$$ is restored to its original state.

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 first line of each test case contains two integers $$$n$$$ and $$$q$$$ ($$$5\le n\le 10^5, 1\le q\le 10^5$$$) — the size of the array and the number of queries.

The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$0\le a_i\lt 2^{30}$$$).

The next $$$q$$$ lines of each test case contain an integer $$$x$$$ ($$$0\le x\lt 2^{30}$$$).

It is guaranteed that the sum of $$$n$$$ and the sum of $$$q$$$ over all test cases do not exceed $$$10^5$$$.

Output

For each test case, print $$$q$$$ integers — the answer to each query.

Example
Input
4
5 1
0 0 1 1 1
1
5 2
6 7 8 9 15
0
1
8 2
102 92 15 19 25 54 62 36
2
1
10 1
56 73 81 23 17 92 50 34 67 78
2
Output
1
9
7
12
31
7
Note

In the first test case, after one transformation, the array becomes $$$[0, 0, 0, 0, 1]$$$. Here $$$\max(a) - \min(a) = 1 - 0 = 1$$$.

In the second test case, the initial array is $$$[6, 7, 8, 9, 15]$$$. Initially, $$$\max(a) - \min(a) = 15 - 6 = 9$$$. After the transformation, the array becomes $$$[1, 1, 6, 7, 8]$$$, where $$$\max(a) - \min(a) = 8 - 1 = 7$$$.