
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
Let \( P \) be a convex polygon each of whose sides touches a circle \( C \) of radius 1. Let \( A \) be the set of points which are a distance 1 or less from \( P \). If \((x, y)\) is a point of \( A \), let \( f(x, y) \) be the number of points in which a unit circle centered at \((x, y)\) intersects \( P \) (so certainly \( f(x, y) \geq 1 \)). What is \( \sup \frac{1}{|A|} \int_A f(x, y) \, dx \, dy \), where the supremum is taken over all possible polygons \(P\)?

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