You are given an array of positive integers nums of length n.

A pair of nonnegative integer arrays (arr1, arr2) of length n is called monotonic if for every index i such that $0 \leq$ i $< n$:

• arr1[i] + arr2[i] $=$ nums[i]

• arr1 is monotonically nondecreasing (that is, arr1[i] $\leq$ arr1[i + 1] for all $0 \leq$ i $< n - 1$)

• arr2 is monotonically nonincreasing (that is, arr2[i] $\geq$ arr2[i + 1] for all $0 \leq$ i $< n - 1$)

Return the total number of valid monotonic pairs. Since the answer may be very large, return it modulo $10^9 + 7$.

Constraints:

• $1 \leq$ n $==$ nums.length $\leq 2000$

• $1 \leq$ nums[i] $\leq 1000$

Now, let’s take a moment to make sure you’ve correctly understood the problem. The quiz below helps you check if you’re solving the correct problem:

We have a game for you to play. Rearrange the logical building blocks to develop a clearer understanding of how to solve this problem.

Implement your solution in the following coding playground.