310891: CF1905F. 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.

5
5
1 2 3 4 5
5
2 1 3 4 5
7
2 1 5 3 7 6 4
6
2 3 5 4 1 6
7
7 6 5 4 3 2 1

3
5
2
3
2

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 $f(p)=5$.