
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
The regular octagon \( A B C D E F G H \) is inscribed in a circle. Points \( P \) and \( Q \) are on the circle, with \( P \) between \( C \) and \( D \), such that \( A P Q \) is an equilateral triangle. It is possible to inscribe a regular \( n \)-sided polygon, one of whose sides is \( P D \), in the circle. What is the value of \( n \)?

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