Let $\operatorname{ GPF}(n)$ be the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$.

Is there a way to prove that the sequence $a_{n+1}=\operatorname{ GPF}(xa_n+y)$, eventually enter a cycle for all positive integers $x,a_0,y>0$?

Is there any set of positive integers $x,a_0,y>0$ such that $a_{n+1}=\operatorname{ GPF}(xa_n+y)\operatorname{ LPF}(xa_n+y)$ diverges?

Where $\operatorname{ LPF}(n)\geq2$ is the *least* prime factor of $n$.