
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 the circle \(x^{2}+y^{2}=4\) and the point \(P(2,1)\), two perpendicular lines are drawn through point \(P\), intersecting the circle at points \(A, B\) and \(C, D\). Point \(A\) lies on the segment \(PB\), and point \(D\) lies on the segment \(PC\). What is the maximum area of the quadrilateral \(ABCD\)?

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