
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
Two circles \(x^2 + y^2 = a^2\) and \(x^2 + y^2 = 9a^2\) intersect a group of parallel lines given by \(y = \pm a \sqrt{3}\) and \(y = \pm 3a \sqrt{3}\), where \(a > 0\). Find the area of the region between these sets of circles and lines, valid for all \(a > 0\).
A) \(\pi a^2\)
B) \(2\pi a^2\)
C) \(4\pi a^2\)
D) \(9\pi a^2\)

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