IOM 2021 Problem 5

There is a safe that can be opened by entering a secret code consisting of $n$ digits, each of them is $0$ or $1$ . Initially, $n$ zeros were entered, and the safe is closed (so, all zeros is not the secret code).

In one attempt, you can enter an arbitrary sequence of $n$ digits, each of them is $0$ or $1$ . If the entered sequence matches the secret code, the safe will open. If the entered sequence matches the secret code in more positions than the previously entered sequence, you will hear a click. In any other cases the safe will remain locked and there will be no click.

Find the smallest number of attempts that is sufficient to open the safe in all cases.