
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
Hawkins, Dustin, and Lucas start playing a game where each begins with $\$2$. A bell rings every $10$ seconds, and each player with money independently chooses one of the other two players at random to give $\$1$. If a player has only $\$1$ left, there is a $\frac{1}{3}$ probability they will keep their money and not give it to anyone. What is the probability that after the bell has rung $2021$ times, each player will still have $\$1$?
A) $\frac{1}{7}$  
B) $\frac{1}{4}$  
C) $\frac{1}{3}$  
D) $\frac{1}{2}$  
E) $\frac{2}{3}$

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