CF2173E Shiro's Mirror Duel

There's no such thing as luck in this world. The victor is decided before the game even starts. — No Game No Life This is an interactive problem. One day, Sora and Shiro feel bored again, so they decide to settle it with a game. At the beginning, Sora gives Shiro a permutation $ ^{\text{∗}} $ $ p_1,p_2,\ldots,p_n $ of length $ n $ . In each operation, Shiro may select two distinct indices $ x $ and $ y $ ( $ 1\le x\ne y\le n $ ). Then Sora flips a fair coin: - With probability $ 0.5 $ , Sora swaps $ p_x $ and $ p_y $ ; - With probability $ 0.5 $ , Sora swaps $ p_{n-x+1} $ and $ p_{n-y+1} $ . After the operation, Sora replies with the actual pair of indices that were swapped, so that Shiro can update her local permutation accordingly. Shiro's goal is to sort the permutation $ p $ in ascending order by using at most $ \lfloor 2.5n+800\rfloor $ operations. Help her! $ ^{\text{∗}} $ A permutation of length $ n $ is an array consisting of $ n $ distinct integers from $ 1 $ to $ n $ in arbitrary order. For example, $ [2,3,1,5,4] $ is a permutation, but $ [1,2,2] $ is not a permutation ( $ 2 $ appears twice in the array), and $ [1,3,4] $ is also not a permutation ( $ n=3 $ but there is $ 4 $ in the array).

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 100 $ ). The description of the test cases follows. The first line of each test case contains a single integer $ n $ ( $ 1\le n\le 4000 $ ) — the length of $ p $ . The second line contains $ n $ integers $ p_1,p_2,\ldots,p_n $ — the elements of $ p $ . It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 2\cdot 10^4 $ . It is guaranteed that there are $ 50 $ tests in this problem.

N/A

In the first test case, $ n=5 $ and the initial permutation is $ [5,1,3,4,2] $ . - We print ? 1 5. The judge replies 1 5, meaning positions $ 1 $ and $ 5 $ are swapped (no mirroring). The array becomes $ [2,1,3,4,5] $ . - We print ? 4 5. The judge replies 2 1, which is the mirror of $ {4,5} $ because with $ n=5 $ we have $ n-4+1=2 $ and $ n-5+1=1 $ . Thus we must swap positions $ 2 $ and $ 1 $ . The array becomes $ [1,2,3,4,5] $ . - The permutation is now sorted, so we print !. The answer line does not count toward the operation limit. In the second test case, the given permutation is already increasing, so we just need to output !.