Erasing Game

Hongjun and Myungwoo play a game with sequences. Hongjun first writes down a sequence $A$ of $N$ natural numbers, chosen however he likes, and Myungwoo writes down a sequence $S$ of the same length in the same way.

The game runs for $N$ rounds. In round $i$, Hongjun has to erase one number from his own sequence $A$ that is not greater than $S_i$. If no such number is left, Hongjun loses. If he finishes all $N$ rounds, Hongjun wins.

Given Myungwoo's sequence $S$, count the sequences $A$ that let Hongjun win when he plays with an optimal strategy. Two sequences with the same numbers in a different order count as different sequences.

The first line contains the length $N$. ($1 \le N \le 200$)

The $i$-th of the next $N$ lines contains $S_i$. ($1 \le S_i \le 10^9$)

Print the number of sequences $A$ that Hongjun can win with, modulo 1,000,000,007.

For $N = 2$ and $S = (1, 2)$, the sequences $A$ that Hongjun can win with are $(1, 1)$, $(1, 2)$, and $(2, 1)$.