
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
Carl is on a vertex of a regular pentagon. Every minute, he randomly selects an adjacent vertex (each with probability \(\frac{1}{2}\)) and walks along the edge to it. What is the probability that after 10 minutes, he ends up where he started?

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