Xor With Smallest Element

Problem

Chef has an array $A$ of length $N$ and an integer $X$.

In one operation, Chef does the following:

• Find the smallest element in the current array. Let this be $S$.
• Next, pick any one index $i$ such that $A_i = S$
• Finally, replace $A_i$ with $A_i \oplus X$

Here $\oplus$ denotes the bitwise XOR operation.

For example, if $A = [1, 1, 2]$ and $X = 4$, then in one move Chef can turn the array into either $[5, 1, 2]$ or $[1, 5, 2]$.

Chef performs this operation exactly $Y$ times. Let $B$ be final array obtained.

Output the array $B$ in sorted order. Note that under this restriction, the output is unique.

Input Format

• The first line of input will contain a single integer $T$, denoting the number of test cases.
• Each test case consists of two lines of input.
  • The first line of each test case contains three space-separated integers $N$, $X$, and $Y$.
  • The second line contains $N$ space-separated integers denoting the array $A$.

Output Format

For each test case, output array $B$ in sorted order.

Constraints

• $1 \leq T \leq 10^4$
• $1 \leq N \leq 10^5$
• $1 \leq Y \leq 10^9$
• $1 \leq A_i , X \leq 2^{30}$
• The sum of $N$ over all test cases won't exceed $10^5$.

Sample 1:

Input

Output

3
1 6 99
9
3 7 44
1 2 3
5 20 6
5 10 15 20 25

15 
3 5 6 
5 20 25 27 30 

Explanation:

Test case $1$: The array is initially $[9]$. Since there is only one element, it will be modified in each step. So,

• After the first operation, the array is $[15]$ (since $9 \oplus 6 = 15$)
• After the second operation, the array is $[9]$ (since $15 \oplus 6 = 9$)
• Continuing the above, it can be verified that after $99$ steps, the array is $[15]$.

Test case $3$: The sequence of steps is as follows:

• Initially, the array is $[5, 10, 15, 20, 25]$
• After operation $1$, it is $[17, 10, 15, 20, 25]$
• After operation $2$, it is $[17, 30, 15, 20, 25]$
• After operation $3$, it is $[17, 30, 27, 20, 25]$
• After operation $4$, it is $[5, 30, 27, 20, 25]$
• After operation $5$, it is $[17, 30, 27, 20, 25]$
• After operation $6$, it is $[5, 30, 27, 20, 25]$

Remember to print the output in sorted order.