Almost Increasing Subsequence

Description:

Input:

The first line of input contains two integers, $$$n$$$ and $$$q$$$ ($$$1 \leq n, q \leq 200\,000$$$)  — the length of the array $$$a$$$ and the number of queries.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \leq a_i \leq 10^9$$$)  — the values of the array $$$a$$$.

Each of the next $$$q$$$ lines contains the description of a query. Each line contains two integers $$$l$$$ and $$$r$$$ ($$$1 \leq l \leq r \leq n$$$)  — the query is about the subarray $$$a_l, a_{l+1}, \dots, a_r$$$.

Output:

For each of the $$$q$$$ queries, print a line containing the length of the longest almost-increasing subsequence of the subarray $$$a_l, a_{l+1}, \dots, a_r$$$.

Sample Input:

9 8
1 2 4 3 3 5 6 2 1
1 3
1 4
2 5
6 6
3 7
7 8
1 8
8 8

Sample Output:

3
4
3
1
4
2
7
1

Note:

In the first query, the subarray is $$$a_1, a_2, a_3 = [1,2,4]$$$. The whole subarray is almost-increasing, so the answer is $$$3$$$.

In the second query, the subarray is $$$a_1, a_2, a_3,a_4 = [1,2,4,3]$$$. The whole subarray is a almost-increasing, because there are no three consecutive elements such that $$$x \geq y \geq z$$$. So the answer is $$$4$$$.

In the third query, the subarray is $$$a_2, a_3, a_4, a_5 = [2, 4, 3, 3]$$$. The whole subarray is not almost-increasing, because the last three elements satisfy $$$4 \geq 3 \geq 3$$$. An almost-increasing subsequence of length $$$3$$$ can be found (for example taking $$$a_2,a_3,a_5 = [2,4,3]$$$ ). So the answer is $$$3$$$.