This is an extension of a previous question of mine (nicely answered by Douglas Zare): Filling a bin with one type of element when uniformly selecting from a set of two (with bias)
Say I fill a multiset by flipping a biased coin and putting the element $A$ in the set if I see heads, and $B$ if I see tails. However, as soon as there are $k$ copies of element $B$ in the multiset, I sequentially select elements with uniform probability (regardless of whether they are $A$ or $B$) and prune them until no more copies of $B$ exist. I then return to flipping my coin and filling the multiset in the aforementioned manner. I halt the process when there exists exactly $N$ copies $A$ in the multiset and no copies of $B$.
If we reach $N$ copies of $A$ in the multiset, and we have $>0$ copies of $B$, we automatically begin the pruning process.
My question here is: does the value of $k$ matter when $k \leq N$? If the value does matter, what value minimizes the number of coin flips required to reach the halt state with exactly $N$ copies of the element $A$ in the multiset and no copies of element $B$?