
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
Amelia has a coin that lands heads with a probability of $\frac{2}{7}$, while Blaine has a coin landing heads with a probability of $\frac{1}{3}$. They take turns tossing their coins; Amelia still starts first. Each player tosses their coin exactly twice each turn. The first player to get heads on both coin tosses within one of their turns wins. If neither player achieves this condition in their first set of turns, they continue into another round of turns under the same rule but summing the total occurrence of double heads from previous rounds. Calculate the probability that Amelia wins.
A) $\frac{1}{4}$
B) $\frac{2}{5}$
C) $\frac{4}{9}$
D) $\frac{5}{7}$
E) $\frac{3}{4}$

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