
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 \( D \) be a disk of radius \( r \). Given \((x, y) \in D\), and \( R > 0 \), let \( a(x, y, R) \) be the length of the arc of the circle centered at \((x, y)\), radius \( R \), which is outside \( D \). Evaluate \(\lim_{R \to 0} R^{-2} \int_D a(x, y, R) \, dx \, dy\).

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