# Reason About Binary Search with One-Call-Delayed Feedback You must find a secret integer in a known inclusive range. A comparison service can report that a guess is too low, too high, or correct. Unlike an ordinary binary-search oracle, the value returned by the current call describes the **previous** guess rather than the guess just submitted. The captured prompt does not define what the first call returns, whether a separate flush operation exists, whether more than one guess may be outstanding, or whether it is safe to submit a duplicate guess solely to receive delayed feedback. It also mentions a second part that was not captured. Explain which protocol details must be clarified before this task has one implementable contract. Then give a binary-search strategy under one explicit, internally consistent delayed-feedback protocol. Track which response belongs to which guess, state the search invariant, explain termination, and analyze comparison and call complexity. Do not reconstruct the missing second part. ### Clarifying Questions to Ask - What does the first call return when there is no previous guess? - Can the client receive the last pending response without submitting another guess? - May multiple guesses be outstanding, and are responses guaranteed to remain in submission order? - Are repeated guesses or a side-effect-free flush call permitted? - What are the inclusive bounds, and is the secret guaranteed to be inside them? ### What a Strong Answer Covers - Recognition that a one-call delay changes the interaction protocol, not the comparison logic itself. - An explicit association between every returned comparison and the earlier guess it evaluates. - A valid startup and completion rule instead of silently treating the first response as current. - The usual binary-search interval invariant and overflow-safe midpoint choice. - A clear call bound for the chosen protocol and an explanation of why other protocol choices may change it. ### Follow-up Questions - If a separate `flush()` returns the pending comparison, how does the call count change? - Can useful guesses be pipelined when each new midpoint depends on the previous comparison? - How should the client handle a response that arrives out of order or names an unexpected guess ID?