Coin Swap VIII

Alice and Bob play a game with a coins. Alice's coin lands heads with probability $0 < p < 1$ of heads per flip, while Bob's coin lands heads with probability $0 < q < 1$ per flip. Alice starts with the coin and flips it. Alice wins when she obtains her first heads, while Bob needs to obtain two heads (not necessarily consecutive) to win. The players repeatedly pass the coin to each other to alternate the flips until someone wins. Find the probability that Alice is the winner when $p = 1/3$ and $q = 1/2$.