KöMaL 2020 K667

Start with a positive integer. In each move, take the half of the number if it is even, or add 1 to the number if it is odd. The sequence of moves terminates if it reaches the number 1. $a)$ Is it true that whatever the starting number is, it is always possible to reach 1 sooner or later (with a finite number of moves)? $b)$ Is it true that at most 30 moves are sufficient to reach 1 if we start from a four-digit number?