Show that : $2, 5, 13, 17, 29, 421, 401, 53, 281,...,\rightarrow \infty$? $a_{n+1}=\operatorname{ GPF}(qa_n+p)$

Question

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

Is there a way to prove that the sequence $a_{n+1}=\operatorname{ GPF}(qa_n+p)$, eventually enter a cycle for all initial values of positive integers $q,a_0,p>1$?

Which my simulations seem to indicate - although the sequence $a_{n+1}=\operatorname{ GPF}(a_n^2+1)$ with $a_0=2$ appears to run off into infinity.