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 python3.

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:
```python
<your code here>
```

Execution time limit: 1.0 seconds
Memory limit: 256.0 MB

# Problem
This is a hard version of the problem. The difference between the easy and hard versions is that in this version, you have to output the lexicographically smallest permutation with the smallest weight.

You are given a permutation $$$p_1, p_2, \ldots, p_n$$$ of integers from $$$1$$$ to $$$n$$$.

Let's define the weight of the permutation $$$q_1, q_2, \ldots, q_n$$$ of integers from $$$1$$$ to $$$n$$$ as $$$$$$|q_1 - p_{q_{2}}| + |q_2 - p_{q_{3}}| + \ldots + |q_{n-1} - p_{q_{n}}| + |q_n - p_{q_{1}}|$$$$$$

You want your permutation to be as lightweight as possible. Among the permutations $$$q$$$ with the smallest possible weight, find the lexicographically smallest.

Permutation $$$a_1, a_2, \ldots, a_n$$$ is lexicographically smaller than permutation $$$b_1, b_2, \ldots, b_n$$$, if there exists some $$$1 \le i \le n$$$ such that $$$a_j = b_j$$$ for all $$$1 \le j < i$$$ and $$$a_i<b_i$$$.

## Input Format
The first line of the input contains a single integer $$$t$$$ ($$$1 \le t \le 100$$$)  — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \le n \le 200$$$)  — the size of the permutation.

The second line of each test case contains $$$n$$$ integers $$$p_1, p_2, \ldots, p_n$$$ ($$$1 \le p_i \le n$$$, all $$$p_i$$$ are distinct)  — the elements of the permutation.

The sum of $$$n$$$ over all test cases doesn't exceed $$$400$$$.

## Output Format
For each test case, output $$$n$$$ integers $$$q_1, q_2, \ldots, q_n$$$ ($$$1 \le q_i \le n$$$, all $$$q_i$$$ are distinct)  — the lexicographically smallest permutation with the smallest weight.

## Examples
```input
3
2
2 1
4
2 3 1 4
5
5 4 3 2 1
```
```output
1 2 
1 3 4 2 
1 3 4 2 5
```

## Note
In the first test case, there are two permutations of length $$$2$$$: $$$(1, 2)$$$ and $$$(2, 1)$$$. Permutation $$$(1, 2)$$$ has weight $$$|1 - p_2| + |2 - p_1| = 0$$$, and the permutation $$$(2, 1)$$$ has the same weight: $$$|2 - p_1| + |1 - p_2| = 0$$$. In this version, you have to output the lexicographically smaller of them  — $$$(1, 2)$$$.

In the second test case, the weight of the permutation $$$(1, 3, 4, 2)$$$ is $$$|1 - p_3| + |3 - p_4| + |4 - p_2| + |2 - p_1| = |1 - 1| + |3 - 4| + |4 - 3| + |2 - 2| = 2$$$. There are no permutations with smaller weights.

In the third test case, the weight of the permutation $$$(1, 3, 4, 2, 5)$$$ is $$$|1 - p_3| + |3 - p_4| + |4 - p_2| + |2 - p_5| + |5 - p_1| = |1 - 3| + |3 - 2| + |4 - 4| + |2 - 1| + |5 - 5| = 4$$$. There are no permutations with smaller weights.

Now solve the problem and return the code.