You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in c++17.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```cpp
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
Monocarp is planning to host a martial arts tournament. There will be three divisions based on weight: lightweight, middleweight and heavyweight. The winner of each division will be determined by a single elimination system.

In particular, that implies that the number of participants in each division should be a power of two. Additionally, each division should have a non-zero amount of participants.

$$$n$$$ participants have registered for the tournament so far, the $$$i$$$-th of them weighs $$$a_i$$$. To split participants into divisions, Monocarp is going to establish two integer weight boundaries $$$x$$$ and $$$y$$$ ($$$x < y$$$).

All participants who weigh strictly less than $$$x$$$ will be considered lightweight. All participants who weigh greater or equal to $$$y$$$ will be considered heavyweight. The remaining participants will be considered middleweight.

It's possible that the distribution doesn't make the number of participants in each division a power of two. It can also lead to empty divisions. To fix the issues, Monocarp can invite an arbitrary number of participants to each division.

Note that Monocarp can't kick out any of the $$$n$$$ participants who have already registered for the tournament.

However, he wants to invite as little extra participants as possible. Help Monocarp to choose $$$x$$$ and $$$y$$$ in such a way that the total amount of extra participants required is as small as possible. Output that amount.

## Input Format
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of testcases.

The first line of each testcase contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of the registered participants.

The second line of each testcase contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le n$$$) — the weights of the registered participants.

The sum of $$$n$$$ over all testcases doesn't exceed $$$2 \cdot 10^5$$$.

## Output Format
For each testcase, print a single integer — the smallest number of extra participants Monocarp is required to invite after he chooses the weight boundaries $$$x$$$ and $$$y$$$.

## Examples
```input
4
4
3 1 2 1
1
1
6
2 2 2 1 1 1
8
6 3 6 3 6 3 6 6
```
```output
0
2
3
2
```

## Note
In the first testcase of the example, Monocarp can choose $$$x=2$$$ and $$$y=3$$$. Lightweight, middleweight and heavyweight divisions will have $$$2$$$, $$$1$$$ and $$$1$$$ participants, respectively. They all are powers of two, so no extra participants are required.

In the second testcase of the example, regardless of the choice of $$$x$$$ and $$$y$$$, one division will have $$$1$$$ participant, the rest will have $$$0$$$. Thus, Monocarp will have to invite $$$1$$$ participant into both of the remaining divisions.

In the third testcase of the example, Monocarp can choose $$$x=1$$$ and $$$y=2$$$. Lightweight, middleweight and heavyweight divisions will have $$$0$$$, $$$3$$$ and $$$3$$$ participants, respectively. So an extra participant is needed in each division.

In the fourth testcase of the example, Monocarp can choose $$$x=8$$$ and $$$y=9$$$. Lightweight, middleweight and heavyweight divisions will have $$$8$$$, $$$0$$$ and $$$0$$$ participants, respectively. Middleweight and heavyweight division need an extra participant each.

Now solve the problem and return the code.