Consider the following algorithm, where f $($n$)$ :$$ n$23$n $+2.$

Input: n in Z$.$

$(1)$ If f $($n$)=0,$ output n and stop. Otherwise, go to step $(2).$

$(2)$ Replace n by f $($n$)\%5$ and go back to step $(1).$

Prove that this algorithm terminates for every choice of input.