DSE 2006 Question 31: Probability and Bayes' theorem

Three players $A$, $B$ and $C$ take turns playing a game as follows. $A$ and $B$ play in the first round. The winner plays $C$ in the second round, while the loser sits out. The winner of the second round plays the person who was sitting out. The game continues in this fashion, with the winner of the current round playing the next round with the person who sits out in the current round. The game ends when a player wins twice in succession; this player is declared the winner of the contest. For any round, each of the two players has probability $1/2$ of winning, regardless of past rounds. The probability that $A$ becomes the winner of the contest is