F. Rudolf and Imbalance
time limit per test
3 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

Rudolf has prepared a set of $n$ problems with complexities $a_1 < a_2 < a_3 < \dots < a_n$. He is not entirely satisfied with the balance, so he wants to add at most one problem to fix it.

For this, Rudolf came up with $m$ models of problems and $k$ functions. The complexity of the $i$-th model is $d_i$, and the complexity of the $j$-th function is $f_j$. To create a problem, he selects values $i$ and $j$ ($1 \le i \le m$, $1 \le j \le k$) and by combining the $i$-th model with the $j$-th function, he obtains a new problem with complexity $d_i + f_j$ (a new element is inserted into the array $a$).

To determine the imbalance of the set, Rudolf sorts the complexities of the problems in ascending order and finds the largest value of $a_i - a_{i - 1}$ ($i > 1$).

What is the minimum value of imbalance that Rudolf can achieve by adding at most one problem, created according to the described rules?

Input

The first line of the input contains a single integer $t$ ($1 \le t \le 10^4$) — the number of testcases.

The first line of each test case contains three integers $n$, $m$, and $k$ ($2 \le n \le 10^5$, $1 \le m, k \le 2 \cdot 10^5$) — the number of prepared problems, the number of models, and the number of functions, respectively.

The second line of each test case contains $n$ integers $a_1, a_2, a_3, \dots a_n$ ($1 \le a_i \le 2 \cdot 10^9$, $a_i < a_{i+1}$) — the complexities of the prepared problems.

The third line of each test case contains $m$ integers $d_1, d_2, d_3, \dots d_m$ ($1 \le d_i \le 10^9$) — the complexities of the models.

The fourth line of each test case contains $k$ integers $f_1, f_2, f_3, \dots f_k$ ($1 \le f_i \le 10^9$) — the complexities of the functions.

It is guaranteed that the sum of $n$ over all testcases does not exceed $10^5$.

It is guaranteed that the sum of $m$ over all testcases does not exceed $2 \cdot 10^5$.

It is guaranteed that the sum of $k$ over all testcases does not exceed $2 \cdot 10^5$.

Output

For each testcase, output a single number — the minimum imbalance that Rudolf can achieve.

Example
Input
75 5 55 10 15 20 2611 14 16 13 816 4 5 3 17 6 51 4 7 10 18 21 222 3 5 7 4 26 8 9 3 27 6 51 4 7 10 18 21 222 3 5 7 4 26 8 13 3 25 6 32 10 13 20 2511 6 10 16 14 56 17 154 2 211 12 14 1519 1410 68 4 23 10 16 18 21 22 29 309 13 16 154 22 4 74 214 15 14 520 1 15 1 12 5 11
Output
5
4
5
8
2
7
11