Sleeping Cows

Farmer John has $N$ $(1 \le N \le 3000)$ cows of various sizes. He originally built each cow a personalized barn, but now some of the cows have outgrown their barns. Specifically, FJ built $N$ barns of sizes $t\_1,t\_2,\ldots,t\_N$, while the cows are now of sizes $s\_1,s\_2,\ldots,s\_N$ ($1\le s\_i,t\_i\le 10^9$).

Every night, the cows go through a ritual of finding a barn to sleep in. Cow $i$ can sleep in barn $j$ if and only if it fits inside the barn ($s\_i\le t\_j$). Each barn can house at most one cow.

A matching of cows to barns is maximal if every cow assigned to a barn fits in that barn, and every unassigned cow is too big for every empty barn left out of the matching.

Compute the number of maximal matchings modulo $10^9 + 7$.

The first line contains $N$.

The second line contains $N$ space-separated integers $s\_1,s\_2,\ldots,s\_N$.

The third line contains $N$ space-separated integers $t\_1,t\_2,\ldots,t\_N$.

Print the number of maximal matchings modulo $10^9 + 7$.

Here is a list of all nine maximal matchings. An ordered pair $(i,j)$ means that cow $i$ is assigned to barn $j$.

(1, 1), (2, 2), (3, 4)
(1, 1), (2, 3), (3, 4)
(1, 1), (2, 4)
(1, 2), (2, 3), (3, 4)
(1, 2), (2, 4)
(1, 3), (2, 2), (3, 4)
(1, 3), (2, 4)
(1, 4), (2, 2)
(1, 4), (2, 3)