Antonio plays a game where he continually flips a fair coin to see the sequence of heads ($H$) and tails ($T$) that he flips. Antonio wins the game if he sees on four consecutive flips the sequence $TTHT$ before he sees the sequence $HTTH$. The probability that Antonio wins the game is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let's think step by step and output the final answer within \boxed{}.
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