
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
Given a regular hexadecagon inscribed in a circle with radius \(r\) inches, calculate the area of the hexadecagon and the ratio of the area of the circle to the area of the hexadecagon.

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