Three players, $A$, $B$, and $C$, play a game by taking turns flipping a coin in the order $A, B, C, A, B, C, \dots$ until someone wins by getting Heads. The coin used has a variable bias:

• For odd-numbered flips (1st, 3rd, 5th, etc.), the probability of Heads is $p = \frac{2}{3}$.
• For even-numbered flips (2nd, 4th, 6th, etc.), the probability of Heads is halved to $\frac{p}{2} = \frac{1}{3}$. What is the probability that player $C$ wins the game?