Smaller Averages

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 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 $[2,2]$.

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 $[2,2,2]$.

The only valid way is to split the first array into $[2],[5,1,3],[2]$ and the second array into $[2],[1,5],[2,2]$.

Problem credits: Alex Liang

Source: USA Computing Olympiad.

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