
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
Three people are playing a passing game where the ball starts with $A$. The probabilities of passing the ball from $A$ to $B$ or $C$, from $B$ to $A$ or $C$, and from $C$ to $A$ or $B$ are all equal. What is the probability that after 2021 passes, the ball will be back with $A$?

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