Polya urn with probability of ball replacement

Question

Suppose we have an urn that initially contains $a$ red and $b$ green balls, where $a$ and $b$ are positive integers. At each discrete time, we select a ball from the urn and then return the ball to the urn along with $c$ new balls of the same color with some probability $p\geq0.5$ or $c$ new balls of the opposite color with some probability $(1-p)$. This seems to be close in spirit to the Bernard Friedman's urn, but it is not the case (as per simulations) that the proportion of red and green balls converge almost surely to $\frac{1}{2}$. How to prove convergence of this process and how to prove the limiting proportion of red balls as a function of the parameters above?