MEX Repetition
Description
You are given an array $a_1,a_2,\ldots, a_n$ of pairwise distinct integers from $0$ to $n$. Consider the following operation:
- consecutively for each $i$ from $1$ to $n$ in this order, replace $a_i$ with $\operatorname{MEX}(a_1, a_2, \ldots, a_n)$.
Here $\operatorname{MEX}$ of a collection of integers $c_1, c_2, \ldots, c_m$ is defined as the smallest non-negative integer $x$ which does not occur in the collection $c$. For example, $\operatorname{MEX}(0, 2, 2, 1, 4) = 3$ and $\operatorname{MEX}(1, 2) = 0$.
Print the array after applying $k$ such operations.
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^5$). The description of the test cases follows.
The first line of each test case contains two integers $n$ and $k$ ($1\le n\le 10^5$, $1\le k\le 10^9$).
The second line contains $n$ pairwise distinct integers $a_1,a_2,\ldots, a_n$ ($0\le a_i\le n$) representing the elements of the array before applying the operations.
It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.
For each test case, print all $n$ elements of the array after applying $k$ operations.
Input
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^5$). The description of the test cases follows.
The first line of each test case contains two integers $n$ and $k$ ($1\le n\le 10^5$, $1\le k\le 10^9$).
The second line contains $n$ pairwise distinct integers $a_1,a_2,\ldots, a_n$ ($0\le a_i\le n$) representing the elements of the array before applying the operations.
It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.
Output
For each test case, print all $n$ elements of the array after applying $k$ operations.
5
1 2
1
3 1
0 1 3
2 2
0 2
5 5
1 2 3 4 5
10 100
5 3 0 4 2 1 6 9 10 8
1
2 0 1
2 1
2 3 4 5 0
7 5 3 0 4 2 1 6 9 10
Note
In the first test case, here is the entire process:
- On the first operation, the array changes from $[1]$ to $[0]$, since $\operatorname{MEX}(1) = 0$.
- On the second operation, the array changes from $[0]$ to $[1]$, since $\operatorname{MEX}(0) = 1$.
Thus, the array becomes $[1]$ after two operations.
In the second test case, the array changes as follows during one operation: $[{\mkern3mu\underline{\mkern-3mu {\bf 0}\mkern-3mu}\mkern3mu}, 1, 3] \rightarrow [2, {\mkern3mu\underline{\mkern-3mu {\bf 1}\mkern-3mu}\mkern3mu}, 3] \rightarrow [2, 0, {\mkern3mu\underline{\mkern-3mu {\bf 3}\mkern-3mu}\mkern3mu}] \rightarrow [2, 0, 1]$.
In the third test case, the array changes as follows during one operation: $[{\mkern3mu\underline{\mkern-3mu {\bf 0}\mkern-3mu}\mkern3mu}, 2] \rightarrow [1, {\mkern3mu\underline{\mkern-3mu {\bf 2}\mkern-3mu}\mkern3mu}] \rightarrow [1, 0]$. And during the second operation: $[{\mkern3mu\underline{\mkern-3mu {\bf 1}\mkern-3mu}\mkern3mu}, 0] \rightarrow [2, {\mkern3mu\underline{\mkern-3mu {\bf 0}\mkern-3mu}\mkern3mu}] \rightarrow [2, 1]$.