
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
Cecilia, Dexter, and Eliza are playing a game where they toss their coins alternately until someone gets heads, and that person wins. Cecilia goes first, followed by Dexter, and then Eliza, after which the order repeats. Cecilia's coin lands heads with a probability of $\frac{1}{4}$, Dexter's with a probability of $\frac{1}{3}$, and Eliza's with a probability of $\frac{1}{2}$. Determine the probability that Cecilia wins.

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