Guess Divisors Count

You are playing one or more independent games against a trusted evaluator. In each game the evaluator has fixed a hidden integer 1 <= X <= 10^9. You do not need to recover X; you need to output an estimate of the number of positive divisors of X.

This is a piped_stdio interactive challenge. At the start of each source testcase the evaluator prints one integer T, the number of games in that testcase. After you answer the T games, the next source testcase, if any, begins immediately with another T. After all Agentics cases are complete, the evaluator prints a single 0 sentinel and closes the session. Your solution may also terminate on EOF.

To ask a query in the current game, print and flush:

0 Q

where 1 <= Q <= 10^18. The evaluator replies with gcd(X, Q). The original source interactor allows at most 100 queries per game.

To finish the current game, print and flush:

1 ans

The answer is accepted if either |ans - d| <= 7 or 1/2 <= ans / d <= 2, where d is the true divisor count of the hidden integer.

Malformed output, an invalid query, exceeding the per-game query limit, EOF before all games are answered, or an answer outside the accepted approximation window receives the original source failure handling. The trusted evaluator alone writes result.json; participant output is never trusted as a result file.

The leaderboard score is the original Testlib ratio scaled to 0..100: (100 - q) / 100, where q is the maximum query count used by any accepted game in the source testcase. Public validation is a tiny deterministic smoke session. Official scoring uses private Frontier-CS hidden integers and judge-owned divisor counts packaged outside Git.