9071: Distance

For the two multisets $A$ and $B$ , In an operation, we can choose one between the following two operations:

1. Update an element aia_iai in the multiset $A$, to ai=ai+1a_i=a_i+1ai=ai+1

2. Update an element bib_ibi in the multiset $B$, to bi=bi+1b_i=b_i+1bi=bi+1

We define C(A,B)C(A,B)C(A,B) as the minimum number of operations to make the multisets $A$ and $B$ identical,meaning that both multisets have the same elements, If there is no way to transform multisets $A$ and $B$ into the same multiset, then C(A,B)=0C(A,B)=0C(A,B)=0.

Now you have two multisets $S$ and $T$, and calc the value of ∑A⊆S∑B⊆TC(A,B)\sum\limits_{A\subseteq S}\sum\limits_{B\subseteq T}C(A,B)A⊆S∑B⊆T∑C(A,B) modulo $998244353$.


Note: Subsets in the multiset are allowed to duplicate values, that is, the number of solutions selected from each of the two sets $S$ and $T$ is (2∣S∣−1)(2∣T∣−1)(2^{|S|}-1)(2^{|T|}-1)(2∣S∣−1)(2∣T∣−1).

The first line contains a positive integer n (1≤n≤2×103)n\ (1\leq n\leq 2\times 10^3)n (1≤n≤2×103),representing the size of the multiset $S$ and the multiset $T$ .
The second line contains a line of $n$ positive integers si (1≤si≤105)s_i \ (1\leq s_i\leq 10^5)si (1≤si≤105), representing the multiset $S$.

Output a line with a integer representing the answer, modulo $998244353$.

A={2} B={1} C(A,B)=1