You are given two sequences $A$ and $B$ of length $N$. Determine the number of different ways to permute the sequence $A$ such that it is lexicographically smaller than the sequence $B$.

A sequence $(X_1, X_2, \dots, X_K)$ is strictly lexicographically smaller than a sequence $(Y_1,Y_2, \dots, Y_K)$, if there exists an index $t$ $(1 \le t \le K)$ such that:

$1. \ X_t < Y_t$

$2. \ X_r = Y_r$ for all $1 \le r < t$

A permutation $X$ of $A$ is considered different from another permutation $Y$ of $A$, if there exists an index $i \ (1 \le i \le N)$ such that $X_i \neq Y_i$. For example:

• If $A = (1,1,1)$, then there is only one different permutation of $A$. 
• If $A = (1,1,2,2)$, then there are $6$ different permutations of $A$.

Input format

• First line: Integer $N \ (1 \le N \le 10^5)$
• Second line: $N$ integers - $A_1, A_2, \dots, A_N \ (1 \le A_i \le 2 \cdot 10^5)$
• Third line: $N$ integers - $B_1, B_2, \dots, B_N \ (1 \le B_i \le 2 \cdot 10^5)$

Output format

Print the remainder of the final answer when divided by $10^9 + 7.$