Detective Nero Wolf is investigating a crime. There are \(80\) people involved in the case. Among them, one is the criminal and another is the only witness to the crime (but the detective does not know who they are).
Each day, the detective may invite any group of these \(80\) people for questioning. If the group contains the witness but does not contain the criminal, then the witness will reveal who the criminal is. Otherwise, nothing happens.
Can the detective guarantee that he solves the case within \(12\) days?