Determining the distribution of results of a simple algorithm
I suspect you won't get a closed form solution for the distribution you're looking for. Think of the seemingly easier problem where $k$ is always chosen to be exactly $2$, and where you get the set $\{1\}$ "for free". This problem is just like asking for a closed-form distribution of the cardinality of the connected component in the Erdos-Renyi graph $G(n,s)$ that contains a specific vertex $v_1$. I'm pretty sure that there's no known closed-form description of this distribution, and that in fact the lower order terms are poorly understood, especially around the critical value $s \sim n \log n$.