Expected Permutation Minimum Product

Given two sequences A and B of length \( n \) satisfying:

• \( A_1 \ge A_2 \ge \cdots \ge A_n \) (i.e. \( A \) is non‐increasing);
• \( A_n \ge \min_{1\le i\le n} B_i \) (i.e. the smallest element in \( A \) is at least the minimum element in \( B \)).

For any permutation \( \pi \) of \( \{1,2,\dots,n\} \), define the value function:

\[ f(\pi)=\prod_{i=1}^n \min\left(A_i, B_{\pi(i)}\right), \]

where each permutation is equally likely. Compute the expected value of \( f(\pi) \) modulo \( 998244353 \), i.e. compute

[ \left(\frac{1}{n!}\sum_{\pi} f(\pi)\right) \bmod 998244353. ]

inputFormat

The input consists of three lines:

• The first line contains a single integer \( n \), the length of the sequences.
• The second line contains \( n \) integers \( A_1, A_2, \dots, A_n \) separated by spaces.
• The third line contains \( n \) integers \( B_1, B_2, \dots, B_n \) separated by spaces.

outputFormat

Output a single integer: the expected value of \( f(\pi) \) modulo \( 998244353 \).

sample

1
5
3

3

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