Permutation Shift
Description
An identity permutation of length $n$ is an array $[1, 2, 3, \dots, n]$.
We performed the following operations to an identity permutation of length $n$:
- firstly, we cyclically shifted it to the right by $k$ positions, where $k$ is unknown to you (the only thing you know is that $0 \le k \le n - 1$). When an array is cyclically shifted to the right by $k$ positions, the resulting array is formed by taking $k$ last elements of the original array (without changing their relative order), and then appending $n - k$ first elements to the right of them (without changing relative order of the first $n - k$ elements as well). For example, if we cyclically shift the identity permutation of length $6$ by $2$ positions, we get the array $[5, 6, 1, 2, 3, 4]$;
- secondly, we performed the following operation at most $m$ times: pick any two elements of the array and swap them.
You are given the values of $n$ and $m$, and the resulting array. Your task is to find all possible values of $k$ in the cyclic shift operation.
The first line contains one integer $t$ ($1 \le t \le 10^5$) — the number of test cases.
Each test case consists of two lines. The first line contains two integers $n$ and $m$ ($3 \le n \le 3 \cdot 10^5$; $0 \le m \le \frac{n}{3}$).
The second line contains $n$ integers $p_1, p_2, \dots, p_n$ ($1 \le p_i \le n$, each integer from $1$ to $n$ appears in this sequence exactly once) — the resulting array.
The sum of $n$ over all test cases does not exceed $3 \cdot 10^5$.
For each test case, print the answer in the following way:
- firstly, print one integer $r$ ($0 \le r \le n$) — the number of possible values of $k$ for the cyclic shift operation;
- secondly, print $r$ integers $k_1, k_2, \dots, k_r$ ($0 \le k_i \le n - 1$) — all possible values of $k$ in increasing order.
Input
The first line contains one integer $t$ ($1 \le t \le 10^5$) — the number of test cases.
Each test case consists of two lines. The first line contains two integers $n$ and $m$ ($3 \le n \le 3 \cdot 10^5$; $0 \le m \le \frac{n}{3}$).
The second line contains $n$ integers $p_1, p_2, \dots, p_n$ ($1 \le p_i \le n$, each integer from $1$ to $n$ appears in this sequence exactly once) — the resulting array.
The sum of $n$ over all test cases does not exceed $3 \cdot 10^5$.
Output
For each test case, print the answer in the following way:
- firstly, print one integer $r$ ($0 \le r \le n$) — the number of possible values of $k$ for the cyclic shift operation;
- secondly, print $r$ integers $k_1, k_2, \dots, k_r$ ($0 \le k_i \le n - 1$) — all possible values of $k$ in increasing order.
Samples
4
4 1
2 3 1 4
3 1
1 2 3
3 1
3 2 1
6 0
1 2 3 4 6 5
1 3
1 0
3 0 1 2
0
Note
Consider the example:
- in the first test case, the only possible value for the cyclic shift is $3$. If we shift $[1, 2, 3, 4]$ by $3$ positions, we get $[2, 3, 4, 1]$. Then we can swap the $3$-rd and the $4$-th elements to get the array $[2, 3, 1, 4]$;
- in the second test case, the only possible value for the cyclic shift is $0$. If we shift $[1, 2, 3]$ by $0$ positions, we get $[1, 2, 3]$. Then we don't change the array at all (we stated that we made at most $1$ swap), so the resulting array stays $[1, 2, 3]$;
- in the third test case, all values from $0$ to $2$ are possible for the cyclic shift:
- if we shift $[1, 2, 3]$ by $0$ positions, we get $[1, 2, 3]$. Then we can swap the $1$-st and the $3$-rd elements to get $[3, 2, 1]$;
- if we shift $[1, 2, 3]$ by $1$ position, we get $[3, 1, 2]$. Then we can swap the $2$-nd and the $3$-rd elements to get $[3, 2, 1]$;
- if we shift $[1, 2, 3]$ by $2$ positions, we get $[2, 3, 1]$. Then we can swap the $1$-st and the $2$-nd elements to get $[3, 2, 1]$;
- in the fourth test case, we stated that we didn't do any swaps after the cyclic shift, but no value of cyclic shift could produce the array $[1, 2, 3, 4, 6, 5]$.