Find the Fake Coin

In front of you are two identical-looking coins, one of which is fair, the other of which comes up Heads 75% of the time. You choose one coin at random and flip it twice, yielding HT.

You will be permitted to perform one more flip, of whichever coin you please, after which you will be asked to guess which coin is the unfair one. Assuming that you choose optimally—that is, you choose which coin to flip so as to maximize the likelihood of eventually identifying the unfair coin—the probability that you will correctly identify the unfair coin can be expressed as \(\frac{m}{n}\), where m and n are positive coprime integers. Find \(m+n\).

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