Divide and Sum
Description
You are given an array $a$ of length $2n$. Consider a partition of array $a$ into two subsequences $p$ and $q$ of length $n$ each (each element of array $a$ should be in exactly one subsequence: either in $p$ or in $q$).
Let's sort $p$ in non-decreasing order, and $q$ in non-increasing order, we can denote the sorted versions by $x$ and $y$, respectively. Then the cost of a partition is defined as $f(p, q) = \sum_{i = 1}^n |x_i - y_i|$.
Find the sum of $f(p, q)$ over all correct partitions of array $a$. Since the answer might be too big, print its remainder modulo $998244353$.
The first line contains a single integer $n$ ($1 \leq n \leq 150\,000$).
The second line contains $2n$ integers $a_1, a_2, \ldots, a_{2n}$ ($1 \leq a_i \leq 10^9$) — elements of array $a$.
Print one integer — the answer to the problem, modulo $998244353$.
Input
The first line contains a single integer $n$ ($1 \leq n \leq 150\,000$).
The second line contains $2n$ integers $a_1, a_2, \ldots, a_{2n}$ ($1 \leq a_i \leq 10^9$) — elements of array $a$.
Output
Print one integer — the answer to the problem, modulo $998244353$.
Samples
1
1 4
6
2
2 1 2 1
12
3
2 2 2 2 2 2
0
5
13 8 35 94 9284 34 54 69 123 846
2588544
Note
Two partitions of an array are considered different if the sets of indices of elements included in the subsequence $p$ are different.
In the first example, there are two correct partitions of the array $a$:
- $p = [1]$, $q = [4]$, then $x = [1]$, $y = [4]$, $f(p, q) = |1 - 4| = 3$;
- $p = [4]$, $q = [1]$, then $x = [4]$, $y = [1]$, $f(p, q) = |4 - 1| = 3$.
In the second example, there are six valid partitions of the array $a$:
- $p = [2, 1]$, $q = [2, 1]$ (elements with indices $1$ and $2$ in the original array are selected in the subsequence $p$);
- $p = [2, 2]$, $q = [1, 1]$;
- $p = [2, 1]$, $q = [1, 2]$ (elements with indices $1$ and $4$ are selected in the subsequence $p$);
- $p = [1, 2]$, $q = [2, 1]$;
- $p = [1, 1]$, $q = [2, 2]$;
- $p = [2, 1]$, $q = [2, 1]$ (elements with indices $3$ and $4$ are selected in the subsequence $p$).