Some 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 $$$a_i \gt a_j$$$ for every $$$j \lt i$$$.

Similarly, suffix maximums are defined, the element $$$a_i$$$ is a suffix maximum if $$$a_i \gt a_j$$$ for every $$$j \gt i$$$.

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 $$$10^9 + 7$$$.

Input

Each 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 $$$p_1 \lt p_2 \lt \ldots \lt p_{m_1}$$$ ($$$1 \le p_i \le n$$$) — the indices of the prefix maximums in increasing order.

The third line of each test case contains $$$m_2$$$ integers $$$s_1 \lt s_2 \lt \ldots \lt s_{m_2}$$$ ($$$1 \le s_i \le n$$$) — 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 $$$2 \cdot 10^5$$$.

Output

For each test case, output a single integer on a separate line — the number of suitable permutations modulo $$$10^9 + 7$$$.

Note

The 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:

$$$[2, 1, 6, 5, 3, 4]$$$
$$$[3, 1, 6, 5, 2, 4]$$$
$$$[3, 2, 6, 5, 1, 4]$$$
$$$[4, 1, 6, 5, 2, 3]$$$
$$$[4, 2, 6, 5, 1, 3]$$$
$$$[4, 3, 6, 5, 1, 2]$$$
$$$[5, 1, 6, 4, 2, 3]$$$
$$$[5, 2, 6, 4, 1, 3]$$$
$$$[5, 3, 6, 4, 1, 2]$$$
$$$[5, 4, 6, 3, 1, 2]$$$

Girl Permutation

Examples