P10282 [USACO24OPEN] Smaller Averages G

Bessie has two arrays of length $N$ ($1 \le N \le 500$). The $i$-th element of the first array is $a_i$ ($1 \le a_i \le 10^6$) and the $i$-th element of the second array is $b_i$ ($1 \le b_i \le 10^6$). Bessie wants to split both arrays into **non-empty** subarrays such that the following is true. 1. Every element belongs in exactly 1 subarray. 2. Both arrays are split into the same number of subarrays. Let the number of subarrays the first and second array are split into be $k$ (i.e. the first array is split into exactly $k$ subarrays and the second array is split into exactly $k$ subarrays). 3. For all $1 \le i \le k$, the average of the $i$-th subarray on the left of the first array is **less than or equal to** the average of the $i$-th subarray on the left of the second array. Count how many ways she can split both arrays into non-empty subarrays while satisfying the constraints modulo $10^9+7$. Two ways are considered different if the number of subarrays are different or if some element belongs in a different subarray.

The first line contains $N$. The next line contains $a_1,a_2,...,a_N$. The next line contains $b_1,b_2,...,b_N$.

Output the number of ways she can split both arrays into non-empty subarrays while satisfying the constraints modulo $10^9+7$.

##### For Sample 1: The two valid ways are: 1. Split the first array into $[1],[2]$ and the second array into $[2],[2]$. 2. Split the first array into $[1,2]$ and the second array into $[2,2]$. ##### For Sample 2: The three valid ways are: 1. Split the first array into $[1,3],[2]$ and the second array into $[2,2],[2]$. 2. Split the first array into $[1,3],[2]$ and the second array into $[2],[2,2]$. 3. Split the first array into $[1,3,2]$ and the second array into $[2,2,2]$. ##### For Sample 3: The only valid way is to split the first array into $[2],[5,1,3],[2]$ and the second array into $[2],[1,5],[2,2]$. #### SCORING: - Inputs 5-6: $N \le 10$ - Inputs 7-9: $N \le 80$ - Inputs 10-17: $N \le 300$ - Inputs 18-20: $N \le 500$