D2. Permutation Weight (Hard Version)time limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputThis 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.InputThe 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.OutputFor 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.ExampleInput
322 142 3 1 455 4 3 2 1Output
1 2 
1 3 4 2 
1 3 4 2 5 
NoteIn 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.
322 142 3 1 455 4 3 2 1
1 2  1 3 4 2  1 3 4 2 5  
['constructive algorithms', 'greedy', '*3500']