Extremum Permutations

Let's consider a permutation P = {p₁, p₂, ..., p_N} of the set of N = {1, 2, 3, ..., N} elements .

P is called a magic set if it satisfies both of the following constraints:

Given a set of K integers, the elements in positions a₁, a₂, ..., a_K are less than their adjacent elements, i.e., p_{a_i-1} > p_{a_i} < p_{a_i+1}
Given a set of L integers, elements in positions b₁, b₂, ..., b_L are greater than their adjacent elements, i.e., p_{b_i-1} < p_{b_i} > p_{b_i+1}

How many such magic sets are there?

Input Format
The first line of input contains three integers N, K, L separated by a single space.
The second line contains K integers, a₁, a₂, ... a_K each separated by single space.
the third line contains L integers, b₁, b₂, ... b_L each separated by single space.

Output Format
Output the answer modulo 1000000007 (10⁹+7).

Constraints
3 <= N <= 5000
1 <= K, L <= 5000
2 <= a_i, b_j <= N-1, where i ∈ [1, K] AND j ∈ [1, L]

Sample Input #00

4 1 1
2
3

Sample Output #00

Explanation #00

Here, N = 4 a₁ = 2 and b₁ = 3. The 5 permutations of {1,2,3,4} that satisfy the condition are

2 1 4 3
3 2 4 1
4 2 3 1
3 1 4 2
4 1 3 2

Sample Input #01

10 2 2
2 4
3 9

Sample Output #01