
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Two three-letter strings,  $aaa$  and  $bbb$ , 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  $a$  when it should have been a  $b$ , or as a  $b$  when it should be an  $a$ . However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let  $S_a$  be the three-letter string received when  $aaa$  is transmitted and let  $S_b$  be the three-letter string received when  $bbb$  is transmitted. Let  $p$  be the probability that  $S_a$  comes before  $S_b$  in alphabetical order. When  $p$  is written as a fraction in lowest terms, what is its numerator?

Present the answer in LaTex format: \boxed{Your answer}
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