CF1553E.Permutation Shift

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.

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 $[1, 2, 3, 4, 6, 5]$ .