Choose the Different Ones!
Contest: Codeforces Round 923 (Div. 3)
Difficulty rating: 1000
Tags: brute force, greedy, math

Given an array $$$a$$$ of $$$n$$$ integers, an array $$$b$$$ of $$$m$$$ integers, and an even number $$$k$$$.

Your task is to determine whether it is possible to choose exactly $$$\frac{k}{2}$$$ elements from both arrays in such a way that among the chosen elements, every integer from $$$1$$$ to $$$k$$$ is included.

For example:

- If $$$a=[2, 3, 8, 5, 6, 5]$$$, $$$b=[1, 3, 4, 10, 5]$$$, $$$k=6$$$, then it is possible to choose elements with values $$$2, 3, 6$$$ from array $$$a$$$ and elements with values $$$1, 4, 5$$$ from array $$$b$$$. In this case, all numbers from $$$1$$$ to $$$k=6$$$ will be included among the chosen elements.
- If $$$a=[2, 3, 4, 5, 6, 5]$$$, $$$b=[1, 3, 8, 10, 3]$$$, $$$k=6$$$, then it is not possible to choose elements in the required way.

Note that you are not required to find a way to choose the elements — your program should only check whether it is possible to choose the elements in the required way.

Input format:
The first line of the input contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. The descriptions of the test cases follow.

The first line of each test case contains three integers $$$n$$$, $$$m$$$, and $$$k$$$ ($$$1 \le n, m \le 2\cdot10^5$$$, $$$2 \le k \le 2 \cdot \min(n, m)$$$, $$$k$$$ is even) — the length of array $$$a$$$, the length of array $$$b$$$, and the number of elements to be chosen, respectively.

The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^6$$$) — the elements of array $$$a$$$.

The third line of each test case contains $$$m$$$ integers $$$b_1, b_2, \dots, b_m$$$ ($$$1 \le b_j \le 10^6$$$) — the elements of array $$$b$$$.

It is guaranteed that the sum of values $$$n$$$ and $$$m$$$ over all test cases in a test does not exceed $$$4 \cdot 10^5$$$.

Output format:
Output $$$t$$$ lines, each of which is the answer to the corresponding test case. As the answer, output "YES" if it is possible to choose $$$\frac{k}{2}$$$ numbers from each array in such a way that among the chosen elements, every integer from $$$1$$$ to $$$k$$$ is included. Otherwise, output "NO".

You can output each letter in any case (lowercase or uppercase). For example, the strings "yEs", "yes", "Yes", and "YES" will be accepted as a positive answer.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
In the first test case of the example, it is possible to choose elements equal to $$$2$$$, $$$3$$$, and $$$6$$$ from array $$$a$$$ and elements equal to $$$1$$$, $$$4$$$, and $$$5$$$ from array $$$b$$$. Thus, all numbers from $$$1$$$ to $$$k=6$$$ are included among the chosen elements.

In the second test case of the example, it can be shown that it is not possible to choose exactly three elements from each array in the required way.

In the third test case of the example, it is possible to choose elements equal to $$$1$$$ and $$$3$$$ from array $$$a$$$ and elements equal to $$$2$$$ and $$$4$$$ from array $$$b$$$. Thus, all numbers from $$$1$$$ to $$$k=4$$$ are included among the chosen elements.

Example input:
6
6 5 6
2 3 8 5 6 5
1 3 4 10 5
6 5 6
2 3 4 5 6 5
1 3 8 10 3
3 3 4
1 3 5
2 4 6
2 5 4
1 4
7 3 4 4 2
1 4 2
2
6 4 4 2
1 5 2
3
2 2 1 4 3

Example output:
YES
NO
YES
YES
NO
NO

Editorial:
Notice that elements with a value greater than $$$k$$$ are not relevant to us. Let's divide the values into three categories:
1. Occurring only in array $$$a$$$;
2. occurring only in array $$$b$$$;
3. occurring in both arrays.
The answer will be NO if any of the following conditions are met:
• the number of values of the first type is greater than $$$\frac{k}{2}$$$ (this implies that we cannot select all such elements);
• the number of values of the second type is greater than $$$\frac{k}{2}$$$ (this implies that we cannot select all such elements);
• the total number of values of all three types is less than $$$k$$$ (this implies that some values do not occur in either of the arrays).
Otherwise, the answer is YES.