HTH vs HH: Which Pattern Appears First?

A fair coin is tossed repeatedly. The game ends the moment either the pattern $HH$ or the pattern $HTH$ appears as consecutive flips -- whichever comes first. What is the probability that $HTH$ appears before $HH$? Model the process as a Markov chain over states that track the relevant suffix of the flip sequence. Set up and solve the system of linear equations exactly.