E. Girl Permutationtime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputSome permutation of length n is guessed.You are given the indices of its prefix maximums and suffix maximums.Recall that a permutation of length k is an array of size k such that each integer from 1 to k occurs exactly once.Prefix maximums are the elements that are the maximum on the prefix ending at that element. More formally, the element a_i is a prefix maximum if a_i > a_j for every j < i.Similarly, suffix maximums are defined, the element a_i is a suffix maximum if a_i > a_j for every j > i.You need to output the number of different permutations that could have been guessed.As this number can be very large, output the answer modulo 10^9 + 7.InputEach test consists of several test cases. The first line contains a single integer t (1 \le t \le 10^4) — the number of test cases. Then follows the description of the test cases.The first line of each test case contains three integers n, m_1 and m_2 (1 \le m_1, m_2 \le n \le 2 \cdot 10^5) — the length of the permutation, the number of prefix maximums, and the number of suffix maximums, respectively.The second line of each test case contains m_1 integers p_1 < p_2 < \ldots < p_{m_1} (1 \le p_i \le n) — the indices of the prefix maximums in increasing order.The third line of each test case contains m_2 integers s_1 < s_2 < \ldots < s_{m_2} (1 \le s_i \le n) — the indices of the suffix maximums in increasing order.It is guaranteed that the sum of the values of n for all test cases does not exceed 2 \cdot 10^5.OutputFor each test case, output a single integer on a separate line — the number of suitable permutations modulo 10^9 + 7.ExampleInput
61 1 1114 2 31 22 3 43 3 11 2 335 3 41 2 32 3 4 520 5 41 2 3 4 1212 13 18 206 2 31 33 4 6Output
1
3
1
0
317580808
10
NoteThe following permutations are suitable for the second set of input data:  [1, 4, 3, 2]  [2, 4, 3, 1]  [3, 4, 2, 1] The following permutations are suitable for the sixth set of input data:  [2, 1, 6, 5, 3, 4]  [3, 1, 6, 5, 2, 4]  [3, 2, 6, 5, 1, 4]  [4, 1, 6, 5, 2, 3]  [4, 2, 6, 5, 1, 3]  [4, 3, 6, 5, 1, 2]  [5, 1, 6, 4, 2, 3]  [5, 2, 6, 4, 1, 3]  [5, 3, 6, 4, 1, 2]  [5, 4, 6, 3, 1, 2] 
61 1 1114 2 31 22 3 43 3 11 2 335 3 41 2 32 3 4 520 5 41 2 3 4 1212 13 18 206 2 31 33 4 6
1 3 1 0 317580808 10 
['combinatorics', 'dp', 'math', 'number theory', '*2200']