
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 and Blaine are playing a modified game where they toss their respective coins. Amelia's coin lands on heads with a probability of $\frac{3}{7}$, and Blaine's lands on heads with a probability of $\frac{1}{3}$. They begin their game only after observing at least one head in a simultaneous toss of both coins. Once the game starts, they toss coins alternately with Amelia starting first, and the player to first toss heads twice consecutively wins the game. What is the probability that Amelia wins the game?
A) $\frac{1}{2}$
B) $\frac{9}{49}$
C) $\frac{2401}{6875}$
D) $\frac{21609}{64328}$

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