C. Bottom-Tier Reversals
constructive algorithms; greedy
You have a permutation: an array \(a = [a_1, a_2, \ldots, a_n]\) of distinct integers from \(1\) to \(n\). The length of the permutation \(n\) is odd.You need to sort the permutation in increasing order.In one step, you can choose any prefix of the permutation with an odd length and reverse it. Formally, if \(a = [a_1, a_2, \ldots, a_n]\), you can choose any odd integer \(p\) between \(1\) and \(n\), inclusive, and set \(a\) to \([a_p, a_{p-1}, \ldots, a_1, a_{p+1}, a_{p+2}, \ldots, a_n]\).Find a way to sort \(a\) using no more than \(\frac{5n}{2}\) reversals of the above kind, or determine that such a way doesn't exist. The number of reversals doesn't have to be minimized.
Each test contains multiple test cases. The first line contains the number of test cases \(t\) (\(1 \le t \le 100\)). Description of the test cases follows.The first line of each test case contains a single integer \(n\) (\(3 \le n \le 2021\); \(n\) is odd) — the length of the permutation.The second line contains \(n\) distinct integers \(a_1, a_2, \ldots, a_n\) (\(1 \le a_i \le n\)) — the permutation itself. It is guaranteed that the sum of \(n\) over all test cases does not exceed \(2021\).
For each test case, if it's impossible to sort the given permutation in at most \(\frac{5n}{2}\) reversals, print a single integer \(-1\).Otherwise, print an integer \(m\) (\(0 \le m \le \frac{5n}{2}\)), denoting the number of reversals in your sequence of steps, followed by \(m\) integers \(p_i\) (\(1 \le p_i \le n\); \(p_i\) is odd), denoting the lengths of the prefixes of \(a\) to be reversed, in chronological order.Note that \(m\) doesn't have to be minimized. If there are multiple answers, print any.
In the first test case, the permutation is already sorted. Any even number of reversals of the length \(3\) prefix doesn't change that fact.In the second test case, after reversing the prefix of length \(3\) the permutation will change to \([5, 4, 3, 2, 1]\), and then after reversing the prefix of length \(5\) the permutation will change to \([1, 2, 3, 4, 5]\).In the third test case, it's impossible to sort the permutation.