On the representation function of a greedy sieve generated by infinite families of quadratic recurrences

Let $\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D}, \mathcal{E}, \mathcal{F}$ be infinite arithmetic progressions of integers. We consider the set of all second-order quadratic recurrence relations: $$s_{n+1} = a s_n^2 + b s_n s_{n-1} + c s_{n-1}^2 + d s_n + e s_{n-1} + f$$ where $(a, b, c, d, e, f) \in \mathcal{A} \times \mathcal{B} \times \mathcal{C} \times \mathcal{D} \times \mathcal{E} \times \mathcal{F}$ and $a > 0$.

We define a Greedy Orbit Partition of $\mathbb{N}$ as follows: Starting from $n=1$, if an integer is "unmarked," it is chosen as a seed $g$ (and subsequently $h$) to initiate a new orbit under one of the maps in our family. An integer $n$ is "marked" each time it appears as an image $s_k$ ($k \ge 2$) of any such orbit.

Let $R(n)$ be the representation function counting the total number of times $n$ is marked across all possible orbits and all possible maps in the infinite family.

The Problem: I am investigating the density of the level sets $S_{\phi} = { n \in \mathbb{N} : R(n) \ge \phi }$.

Numerical Conjecture: Based on large-scale simulations (up to $n=10^9$ with large finite subsets of the coefficient space), I observe a possible Phase Transition in the convergence of the sum of reciprocals:

The set $S_1$ is Large (the sum of reciprocals $\sum_{n \in S_1} \frac{1}{n}$ appears to diverge). There exists a finite threshold $\phi_c$ such that for $\phi > \phi_c$, the set $S_{\phi}$ is Small (the sum of reciprocals converges). My Questions:

Given that the family of maps is infinite, what determines the finiteness of $R(n)$? Is $R(n)$ almost surely finite for all $n$? Does the quadratic growth of the maps ($s_n^2$) guarantee a "Small" sum of reciprocals for $S_{\phi}$ even when the number of maps is infinite, or would I need to shift to geometric progressions for the coefficients to guarantee this dichotomy? How does the "Greedy" selection of seeds (avoiding marked seeds) affect the asymptotic density of the marked set compared to a standard union of ranges?