
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
Let's assume Jackie and Phil have now four fair coins and a third coin that comes up heads with probability $\frac{3}{5}$. Both Jackie and Phil take turns to flip these five coins. Find the smallest integer value $x$ so that the probability that Jackie gets the same number of heads as Phil is greater than $\frac{1}{x}$.

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