Two three-letter strings, and , are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an when it should have been a , or as a when it should be an . However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let be the three-letter string received when is transmitted and let be the three-letter string received when is transmitted. Let be the probability that comes before in alphabetical order. When is written as a fraction in lowest terms, what is its numerator?
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