E. So Mean
interactive; math
This problem is interactive.We have hidden a permutation \(p_1, p_2, \dots, p_n\) of numbers from \(1\) to \(n\) from you, where \(n\) is even. You can try to guess it using the following queries:\(?\) \(k\) \(a_1\) \(a_2\) \(\dots\) \(a_k\).In response, you will learn if the average of elements with indexes \(a_1, a_2, \dots, a_k\) is an integer. In other words, you will receive \(1\) if \(\frac{p_{a_1} + p_{a_2} + \dots + p_{a_k}}{k}\) is integer, and \(0\) otherwise. You have to guess the permutation. You can ask not more than \(18n\) queries.Note that permutations \([p_1, p_2, \dots, p_k]\) and \([n + 1 - p_1, n + 1 - p_2, \dots, n + 1 - p_k]\) are indistinguishable. Therefore, you are guaranteed that \(p_1 \le \frac{n}{2}\).Note that the permutation \(p\) is fixed before the start of the interaction and doesn't depend on your queries. In other words, interactor is not adaptive.Note that you don't have to minimize the number of queries.
The first line contains a single integer \(n\) (\(2 \le n \le 800\), \(n\) is even).