Absolute Beauty
Description
Kirill has two integer arrays $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ of length $n$. He defines the absolute beauty of the array $b$ as $$\sum_{i=1}^{n} |a_i - b_i|.$$ Here, $|x|$ denotes the absolute value of $x$.
Kirill can perform the following operation at most once:
- select two indices $i$ and $j$ ($1 \leq i < j \leq n$) and swap the values of $b_i$ and $b_j$.
Help him find the maximum possible absolute beauty of the array $b$ after performing at most one swap.
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \leq t \leq 10\,000$). The description of test cases follows.
The first line of each test case contains a single integer $n$ ($2\leq n\leq 2\cdot 10^5$) — the length of the arrays $a$ and $b$.
The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1\leq a_i\leq 10^9$) — the array $a$.
The third line of each test case contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1\leq b_i\leq 10^9$) — the array $b$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2\cdot 10^5$.
For each test case, output one integer — the maximum possible absolute beauty of the array $b$ after no more than one swap.
Input
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \leq t \leq 10\,000$). The description of test cases follows.
The first line of each test case contains a single integer $n$ ($2\leq n\leq 2\cdot 10^5$) — the length of the arrays $a$ and $b$.
The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1\leq a_i\leq 10^9$) — the array $a$.
The third line of each test case contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1\leq b_i\leq 10^9$) — the array $b$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2\cdot 10^5$.
Output
For each test case, output one integer — the maximum possible absolute beauty of the array $b$ after no more than one swap.
6
3
1 3 5
3 3 3
2
1 2
1 2
2
1 2
2 1
4
1 2 3 4
5 6 7 8
10
1 8 2 5 3 5 3 1 1 3
2 9 2 4 8 2 3 5 3 1
3
47326 6958 358653
3587 35863 59474
4
2
2
16
31
419045
Note
In the first test case, each of the possible swaps does not change the array $b$.
In the second test case, the absolute beauty of the array $b$ without performing the swap is $|1-1| + |2-2| = 0$. After swapping the first and the second element in the array $b$, the absolute beauty becomes $|1-2| + |2-1| = 2$. These are all the possible outcomes, hence the answer is $2$.
In the third test case, it is optimal for Kirill to not perform the swap. Similarly to the previous test case, the answer is $2$.
In the fourth test case, no matter what Kirill does, the absolute beauty of $b$ remains equal to $16$.