Decinc Dividing
Description
Let's call an array $a$ of $m$ integers $a_1, a_2, \ldots, a_m$ Decinc if $a$ can be made increasing by removing a decreasing subsequence (possibly empty) from it.
- For example, if $a = [3, 2, 4, 1, 5]$, we can remove the decreasing subsequence $[a_1, a_4]$ from $a$ and obtain $a = [2, 4, 5]$, which is increasing.
You are given a permutation $p$ of numbers from $1$ to $n$. Find the number of pairs of integers $(l, r)$ with $1 \le l \le r \le n$ such that $p[l \ldots r]$ (the subarray of $p$ from $l$ to $r$) is a Decinc array.
The first line contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the size of $p$.
The second line contains $n$ integers $p_1, p_2, \ldots, p_n$ ($1 \le p_i \le n$, all $p_i$ are distinct) — elements of the permutation.
Output the number of pairs of integers $(l, r)$ such that $p[l \ldots r]$ (the subarray of $p$ from $l$ to $r$) is a Decinc array. $(1 \le l \le r \le n)$
Input
The first line contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the size of $p$.
The second line contains $n$ integers $p_1, p_2, \ldots, p_n$ ($1 \le p_i \le n$, all $p_i$ are distinct) — elements of the permutation.
Output
Output the number of pairs of integers $(l, r)$ such that $p[l \ldots r]$ (the subarray of $p$ from $l$ to $r$) is a Decinc array. $(1 \le l \le r \le n)$
Samples
3
2 3 1
6
6
4 5 2 6 1 3
19
10
7 10 1 8 3 9 2 4 6 5
39
Note
In the first sample, all subarrays are Decinc.
In the second sample, all subarrays except $p[1 \ldots 6]$ and $p[2 \ldots 6]$ are Decinc.