
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
Jackie and Phil have two fair coins and a third coin that comes up heads with probability  $ \frac47$ . Jackie flips the three coins, and then Phil flips the three coins. Let  $ \frac{m}{n}$  be the probability that Jackie gets the same number of heads as Phil, where  $ m$  and  $ n$  are relatively prime positive integers. Find  $ m \plus{} n$ .

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