
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
Alice, Bob, and Charlie each flip a fair coin repeatedly until they each flip heads. In a separate event, three more people, Dave, Eve, and Frank, each flip a biased coin (with a probability of $\frac{1}{3}$ of getting heads) until they first flip heads. Determine the probability that both groups will stop flipping their coins on the same round.

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