
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
Given that Charlie, Amelia, and Blaine play a game where Charlie goes third with a coin landing heads at $\frac{1}{5}$, Amelia goes second with a coin that lands heads with probability $\frac{1}{4}$, and Blaine goes first with a coin that lands heads with probability $\frac{1}{3}$, find the difference between the denominator and numerator of the probability that Blaine wins the game.

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