A controlled finite Markov chain with arbitrary set of decisions

We consider a controlled Markov chain with a finite set $S$ of states $s$ and an arbitrary set $A$ of decisions $a$ and with the optimality criterion of the form
$$ \mathbf E^\pi\biggl[\sum_{n=1}^\tau r(s_n,a_n)+c(s_\tau,a_\tau)\biggr], $$
where the stopping moment $\tau$ does not depend on $(s_n,a_n);n\ge1)$ and has the geometric distribution.
Sufficient conditions for the existence of $(k,\varepsilon)$-optimal policies are found.