Race to Three Heads

Two players take turns flipping a fair coin. The first player to accumulate 3 consecutive heads on their own flips wins. Players alternate turns, flipping once per turn. You go first and flip heads on your very first flip. What is the probability that you win the game? Clarifications: - Each player's "streak" only counts consecutive heads on their own turns. Your opponent's flips do not break your streak, and vice versa. - If you flip tails, your streak resets to 0. If you flip heads, your streak increments by 1. - The game ends the moment either player reaches a streak of 3 heads.