CF1685D2.Permutation Weight (Hard Version)

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 $1 \le j \lt i$ and $a_i \lt b_i$.

这是一个该问题的困难版本。简单版本与困难版本的区别在于：在本版本中，你需要输出权重最小的字典序最小的排列。

给你一个 $1$ 到 $n$ 的整数排列 $p_1, p_2, \ldots, p_n$。

我们定义 $1$ 到 $n$ 的整数排列 $q_1, q_2, \ldots, q_n$ 的权重为

$$|q\_1 - p\_{q\_{2}}| + |q\_2 - p\_{q\_{3}}| + \ldots + |q\_{n-1} - p\_{q\_{n}}| + |q\_n - p\_{q\_{1}}|$$

你希望你的排列尽可能轻（即权重尽可能小）。在所有具有最小可能权重的排列 $q$ 中，请找出字典序最小的一个。

排列 $a_1, a_2, \ldots, a_n$ 字典序小于排列 $b_1, b_2, \ldots, b_n$，当且仅当存在某个 $1 \le i \le n$，使得对所有 $1 \le j < i$ 均有 $a_j = b_j$，且 $a_i < b_i$。

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 $400$.

输入的第一行包含一个整数 $t$（$1 \le t \le 100$），表示测试用例的数量。随后是各测试用例的描述。

每个测试用例的第一行包含一个整数 $n$（$2 \le n \le 200$），表示排列的长度。

每个测试用例的第二行包含 $n$ 个整数 $p_1, p_2, \ldots, p_n$（$1 \le p_i \le n$，且所有 $p_i$ 互不相同），表示该排列的各个元素。

所有测试用例的 $n$ 值之和不超过 $400$。

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.

对于每个测试用例，输出 $n$ 个整数 $q_1, q_2, \ldots, q_n$（其中 $1 \le q_i \le n$，且所有 $q_i$ 互不相同）——即具有最小权值的字典序最小的排列。

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.

在第一个测试用例中，长度为 $2$ 的排列有两个：$(1, 2)$ 和 $(2, 1)$。排列 $(1, 2)$ 的权重为 $|1 - p_2| + |2 - p_1| = 0$，而排列 $(2, 1)$ 具有相同的权重：$|2 - p_1| + |1 - p_2| = 0$。在此版本中，你需要输出字典序更小的那个——即 $(1, 2)$。

在第二个测试用例中，排列 $(1, 3, 4, 2)$ 的权重为 $|1 - p_3| + |3 - p_4| + |4 - p_2| + |2 - p_1| = |1 - 1| + |3 - 4| + |4 - 3| + |2 - 2| = 2$。不存在权重更小的排列。

在第三个测试用例中，排列 $(1, 3, 4, 2, 5)$ 的权重为 $|1 - p_3| + |3 - p_4| + |4 - p_2| + |2 - p_5| + |5 - p_1| = |1 - 3| + |3 - 2| + |4 - 4| + |2 - 1| + |5 - 5| = 4$。不存在权重更小的排列。