A16269. Field Should Not Be Empty

You are given a permutation $^{\dagger}$ $p$ of length $n$ .

We call index $x$ good if for all $y < x$ it holds that $p_y < p_x$ and for all $y > x$ it holds that $p_y > p_x$ . We call $f(p)$ the number of good indices in $p$ .

You can perform the following operation: pick $2$ distinct indices $i$ and $j$ and swap elements $p_i$ and $p_j$ .

Find the maximum value of $f(p)$ after applying the aforementioned operation exactly once.

$^{\dagger}$ 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).

Each test consists of multiple test cases. The first line of contains a single integer $t$ ( $1 \le t \le 2 \cdot 10^4$ ) — the number of test cases. The description of the test cases follows.

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

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

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

For each test case, output a single integer — the maximum value of $f(p)$ after performing the operation exactly once.

In the first test case, $p = [1,2,3,4,5]$ and $f(p)=5$ which is already maximum possible. But must perform the operation anyway. We can get $f(p)=3$ by choosing $i=1$ and $j=2$ which makes $p = [2,1,3,4,5]$ .

In the second test case, we can transform $p$ into $[1,2,3,4,5]$ by choosing $i=1$ and $j=2$ . Thus $f(p)=5$ .