Bracket Sequence Recovery

This is an interactive challenge. A hidden bracket sequence s of length n contains at least one ( and at least one ). You can query subsequences by index and must recover the whole string.

The original Frontier-CS problem was interactive. This Agentics migration keeps the source interactor.cpp protocol in a piped_stdio session.

Input

For each source case, the evaluator writes one integer n. Public validation uses a tiny smoke case; official evaluation uses private Frontier-CS cases.

An Agentics session may contain more than one original Frontier-CS case. After you answer one case, keep reading stdin. If another positive n arrives, solve that case with the same protocol. When the evaluator writes 0, the session is complete and your program should exit. EOF before the terminal marker should also be treated as the end of interaction.

Output

To ask a query, output:

0 k i_1 i_2 ... i_k

Then flush stdout and read one integer response from stdin. The response is f(t), the number of non-empty regular bracket substrings in t = s[i_1]s[i_2]...s[i_k]. Indices may repeat. Each query must satisfy 1 <= k <= 1000 and 1 <= i_j <= n.

To guess the current hidden sequence, output:

1 s

The guess does not count as a query. For a multi-case Agentics session, continue reading after each guess until the evaluator sends terminal n = 0.

Scoring

The source interactor allows at most 200 queries per case. If the final string is correct, the source ratio is (200 - q) / 200, where q is the number of queries. Protocol errors, too many queries, invalid indices, or a wrong final string receive zero for the affected case. Agentics reports the average source ratio as score from 0 to 100.

Solution Interface

Submit a zip_project solution with an agentics.solution.json manifest. The manifest-declared run command is connected to the trusted interactive evaluator through stdin/stdout. Network access is disabled.