CF1838B.Minimize Permutation Subarrays

You are given a permutation $p$ of size $n$ . You want to minimize the number of subarrays of $p$ that are permutations. In order to do so, you must perform the following operation exactly once:

• Select integers $i$ , $j$ , where $1 \le i, j \le n$ , then
• Swap $p_i$ and $p_j$ .

For example, if $p = [5, 1, 4, 2, 3]$ and we choose $i = 2$ , $j = 3$ , the resulting array will be $[5, 4, 1, 2, 3]$ . If instead we choose $i = j = 5$ , the resulting array will be $[5, 1, 4, 2, 3]$ .

Which choice of $i$ and $j$ will minimize the number of subarrays that are permutations?

A permutation of length $n$ is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ( $2$ appears twice in the array), and $[1,3,4]$ is also not a permutation ( $n=3$ but there is $4$ in the array).

An array $a$ is a subarray of an array $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.

The first line of the input contains a single integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $n$ ( $3 \le n \le 2\cdot 10^5$ ) — the size of the permutation.

The next line of each test case contains $n$ integers $p_1, p_2, \ldots p_n$ ( $1 \le p_i \le n$ , all $p_i$ are distinct) — the elements of the permutation $p$ .

It is guaranteed that the sum of $n$ over all test cases does not exceed $2\cdot 10^5$ .

For each test case, output two integers $i$ and $j$ ( $1 \le i, j \le n$ ) — the indices to swap in $p$ .

If there are multiple solutions, print any of them.

For the first test case, there are four possible arrays after the swap:

• If we swap $p_1$ and $p_2$ , we get the array $[2, 1, 3]$ , which has 3 subarrays that are permutations ( $[1]$ , $[2, 1]$ , $[2, 1, 3]$ ).
• If we swap $p_1$ and $p_3$ , we get the array $[3, 2, 1]$ , which has 3 subarrays that are permutations ( $[1]$ , $[2, 1]$ , $[3, 2, 1]$ ).
• If we swap $p_2$ and $p_3$ , we get the array $[1, 3, 2]$ , which has 2 subarrays that are permutations ( $[1]$ , $[1, 3, 2]$ ).
• If we swap any element with itself, we get the array $[1, 2, 3]$ , which has 3 subarrays that are permutations ( $[1]$ , $[1, 2]$ , $[1, 2, 3]$ ).

So the best swap to make is positions $2$ and $3$ .For the third sample case, after we swap elements at positions $2$ and $5$ , the resulting array is $[1, 4, 2, 5, 3]$ . The only subarrays that are permutations are $[1]$ and $[1, 4, 2, 5, 3]$ . We can show that this is minimal.