
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
Samantha and Alex each have one fair coin and two more coins which are biased. One coin comes up heads with a probability of $\frac{3}{5}$, and the other has heads with probability $\frac{2}{3}$. They both flip their set of three coins. Let $\frac{m}{n}$ be the probability that Samantha gets more heads than Alex, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

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