A16341 | Neutral Tonality

You are given an array $a$ consisting of $n$ integers, as well as an array $b$ consisting of $m$ integers.

Let $\text{LIS}(c)$ denote the length of the [longest increasing subsequence](https://en.wikipedia.org/wiki/Longest_increasing_subsequence) of array $c$ . For example, $\text{LIS}([2, \underline{1}, 1, \underline{3}])$ = $2$ , $\text{LIS}([\underline{1}, \underline{7}, \underline{9}])$ = $3$ , $\text{LIS}([3, \underline{1}, \underline{2}, \underline{4}])$ = $3$ .

You need to insert the numbers $b_1, b_2, \ldots, b_m$ into the array $a$ , at any positions, in any order. Let the resulting array be $c_1, c_2, \ldots, c_{n+m}$ . You need to choose the positions for insertion in order to minimize $\text{LIS}(c)$ .

Formally, you need to find an array $c_1, c_2, \ldots, c_{n+m}$ that simultaneously satisfies the following conditions:

- The array $a_1, a_2, \ldots, a_n$ is a subsequence of the array $c_1, c_2, \ldots, c_{n+m}$ .
- The array $c_1, c_2, \ldots, c_{n+m}$ consists of the numbers $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_m$ , possibly rearranged.
- The value of $\text{LIS}(c)$ is the minimum possible among all suitable arrays $c$ .

Each test contains multiple test cases. The first line contains a single integer $t$ $(1 \leq t \leq 10^4)$ — the number of test cases. The description of the test cases follows.

The first line of each test case contains two integers $n, m$ $(1 \leq n \leq 2 \cdot 10^5, 1 \leq m \leq 2 \cdot 10^5)$ — the length of array $a$ and the length of array $b$ .

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ $(1 \leq a_i \leq 10^9)$ — the elements of the array $a$ .

The third line of each test case contains $m$ integers $b_1, b_2, \ldots, b_m$ $(1 \leq b_i \leq 10^9)$ — the elements of the array $b$ .

It is guaranteed that the sum of $n$ over all test cases does not exceed $2\cdot 10^5$ , and the sum of $m$ over all test cases does not exceed $2\cdot 10^5$ .

For each test case, output $n + m$ numbers — the elements of the final array $c_1, c_2, \ldots, c_{n+m}$ , obtained after the insertion, such that the value of $\text{LIS}(c)$ is minimized. If there are several answers, you can output any of them.