
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
When flipped, coin A shows heads  $\textstyle\frac{1}{3}$  of the time, coin B shows heads  $\textstyle\frac{1}{2}$  of the time, and coin C shows heads  $\textstyle\frac{2}{3}$  of the time. Anna selects one of the coins at random and flips it four times, yielding three heads and one tail. The probability that Anna flipped coin A can be expressed as  $\textstyle\frac{p}{q}$  for relatively prime positive integers  $p$  and  $q$ . Compute  $p + q$ .

*Proposed by Eugene Chen*

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