
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
Suppose \( A, B, C \), and \( D \) are four circles of radius \( r > 0 \) centered at the points \((0, r)\), \((r, 0)\), \((0, -r)\), and \((-r, 0)\) in the plane. Let \( O \) be a circle centered at \((0, 0)\) with radius \( 2r \). In terms of \( r \), what is the area of the union of circles \( A, B, C \), and \( D \) subtracted by the area of circle \( O \) that is not contained in the union of \( A, B, C \), and \( D \)?

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