### Problem Statement

A permutation A[0], A[1], ..., A[N-1] is a sequence containing each integer between 0 and N-1, inclusive, exactly once. Each permutation A of length N has a corresponding child array B of the same length, where B is defined as follows:
1. B[0] = 0
2. B[i] = A[B[i-1]], for every i between 1 and N-1, inclusive.
A permutation is considered perfect if its child array is also a permutation.

Below are given all permutations for N=3 with their child arrays. Note that for two of these permutations ({1, 2, 0} and {2, 0, 1}) the child array is also a permutation, so these two permutations are perfect.
```Permutation		Child array
{0, 1, 2}		{0, 0, 0}
{0, 2, 1}		{0, 0, 0}
{1, 0, 2}		{0, 1, 0}
{1, 2, 0}		{0, 1, 2}
{2, 0, 1}		{0, 2, 1}
{2, 1, 0}		{0, 2, 0}
```
You are given a int[] P containing a permutation of length N. Find a perfect permutation Q of the same length such that the difference between P and Q is as small as possible. The difference between P and Q is the number of indices i for which P[i] and Q[i] are different. If there are several such permutations Q, return the one among them that has the lexicographically smallest child array.

### Definition

 Class: PerfectPermutationHard Method: reorder Parameters: int[] Returns: int[] Method signature: int[] reorder(int[] P) (be sure your method is public)

### Notes

-int[] A comes before int[] B (with the same length) lexicographically if A has a smaller integer at the first position where the arrays differ.

### Constraints

-P will contain between 1 and 50 elements, inclusive.
-P will contain each integer between 0 and N-1, inclusive, exactly once, where N is the number of elements in P.

### Examples

0)

 `{2, 0, 1}`
`Returns: {2, 0, 1 }`
1)

 `{4, 0, 5, 2, 1, 3}`
`Returns: {2, 0, 5, 4, 1, 3 }`
 Here the smallest possible difference between P and Q is 2. There are 9 possible choices for Q: {2,0,5,4,1,3}, {3,0,5,2,1,4}, {4,0,1,2,5,3}, {4,0,5,1,2,3}, {4,0,5,2,3,1}, {4,2,5,0,1,3}, {4,3,5,2,1,0}, {4,5,0,2,1,3} and {5,0,4,2,1,3}. Among them, {2,0,5,4,1,3} has the lexicographically smallest child array (this array is {0,2,5,3,4,1}).
2)

 `{2, 7, 3, 0, 6, 4, 5, 1}`
`Returns: {1, 7, 3, 0, 6, 2, 5, 4 }`
3)

 `{11, 8, 10, 1, 5, 4, 0, 7, 3, 9, 12, 6, 2}`
`Returns: {1, 8, 10, 2, 5, 7, 0, 9, 3, 11, 12, 6, 4 }`
4)

 `{0, 1, 4, 2, 3, 5}`
`Returns: {1, 2, 4, 5, 3, 0 }`
5)

 `{0, 2, 6, 5, 7, 3, 1, 4}`
`Returns: {1, 2, 6, 5, 7, 4, 3, 0 }`

#### Problem url:

http://www.topcoder.com/stat?c=problem_statement&pm=10469

#### Problem stats url:

http://www.topcoder.com/tc?module=ProblemDetail&rd=13749&pm=10469

giolekva

#### Testers:

PabloGilberto , lovro , ivan_metelsky

#### Problem categories:

Greedy, Simple Math, Simple Search, Iteration