Coin Swap II: Equal Winning Probability

Alice and Bob play a game with a biased coin that shows heads with probability $0 < p \leq 1$. Alice goes first. The rules are asymmetric: - On Alice's turn, she flips the coin once. If she gets heads, she wins. If tails, she passes the coin to Bob. - On Bob's turn, he flips the coin twice. If either flip is heads, he wins. If both are tails, he passes the coin back to Alice. The game continues until someone flips heads. Find the value of $p$ such that Alice and Bob are equally likely to win. Round to the nearest thousandth.