
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 Carol each start with $3. Every 20 seconds, a bell rings, at which time each of the players who has an odd amount of money must give $1 to one of the other two players randomly. What is the probability that after the bell has rung 100 times, each player has exactly $3?
A) $\frac{1}{3}\quad$ B) $\frac{1}{2}\quad$ C) $\frac{9}{13}\quad$ D) $\frac{8}{13}\quad$ E) $\frac{3}{4}$

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