CF1682C.LIS or Reverse LIS?

You are given an array $a$ of $n$ positive integers.

Let $\text{LIS}(a)$ denote the length of longest strictly increasing subsequence of $a$ . For example,

• $\text{LIS}([2, \underline{1}, 1, \underline{3}])$ = $2$ .
• $\text{LIS}([\underline{3}, \underline{5}, \underline{10}, \underline{20}])$ = $4$ .
• $\text{LIS}([3, \underline{1}, \underline{2}, \underline{4}])$ = $3$ .

We define array $a'$ as the array obtained after reversing the array $a$ i.e. $a' = [a_n, a_{n-1}, \ldots , a_1]$ .

The beauty of array $a$ is defined as $min(\text{LIS}(a),\text{LIS}(a'))$ .

Your task is to determine the maximum possible beauty of the array $a$ if you can rearrange the array $a$ arbitrarily.

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

The first line of each test case contains a single integer $n$ $(1 \leq n \leq 2\cdot 10^5)$ — the length of array $a$ .

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 $a$ .

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

For each test case, output a single integer — the maximum possible beauty of $a$ after rearranging its elements arbitrarily.

In the first test case, $a$ = $[6, 6, 6]$ and $a'$ = $[6, 6, 6]$ . $\text{LIS}(a) = \text{LIS}(a')$ = $1$ . Hence the beauty is $min(1, 1) = 1$ .

In the second test case, $a$ can be rearranged to $[2, 5, 4, 5, 4, 2]$ . Then $a'$ = $[2, 4, 5, 4, 5, 2]$ . $\text{LIS}(a) = \text{LIS}(a') = 3$ . Hence the beauty is $3$ and it can be shown that this is the maximum possible beauty.

In the third test case, $a$ can be rearranged to $[1, 2, 3, 2]$ . Then $a'$ = $[2, 3, 2, 1]$ . $\text{LIS}(a) = 3$ , $\text{LIS}(a') = 2$ . Hence the beauty is $min(3, 2) = 2$ and it can be shown that $2$ is the maximum possible beauty.