KöMaL 2021 B5186

Al and Bill are playing the following game. They agree on a fixed number $n \ge 3$, and then Al thinks of a number from the set $\{1,2,\ldots,n \}$. Now Bill can guess the number. He will only get yes or no answers. If the answer is yes, the game terminates. If the answer is no, Al will change the number: either increases or reduces it by 1, but the number must remain positive (it is allowed to go beyond $n$ though). Then Bill can guess again, trying to hit the new number. The procedure is repeated until finally Bill gets the number. Prove that Bill has a strategy to end the game with at most $(3n-5)$ guesses.