On terms in a dynamical divisibility sequence having a fixed G.C.D with their index

Let $F(x)$ be an integer coefficient polynomial with degree at least $2.$ Define the sequence $a_n$ by $a_n=F(a_{n-1})$ for all $n\ge 1$ and $a_0=0.$ Let $\mathscr{B}_{F,G,k}$ be the set of all positive integers $n$ such that $k\mid \gcd(G(n),a_n)$ and if $p\mid \gcd(G(n),a_n)$ for some $p$, then $p\mid k.$ And $\mathscr{A}_{F,G,k}$ be the subset of $\mathscr{B}_{F,G,k}$ such that $\mathscr{A}_{F,G,k}=\{n>0 : \gcd(G(n),a_n)=k\}.$ In this article, we explain the asymptotic density of $\mathscr{A}_{F,G,k}$ and $\mathscr{B}_{F,G,k}$ for a class of $(F,G)$ and also compute the explicit density of $\mathscr{A}_{F,G,k}$ and $\mathscr{B}_{F,G,k}$ for $G(x)=x.$