Coin Swap XII

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. The first player to flip $2$ heads (not necessarily consecutive) wins. The players repeatedly alternate the flips until someone wins. Let $\varphi(p,q)$ represent the probability that Alice wins the game given the probabilities $p$ and $q$ above. If $q$ is fixed, find $\displaystyle \lim_{p \rightarrow 0}\varphi(p,q)$ when $q = 1/2$