
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
In a plane, two circles are constructed such that each passes through the center of the other. Points \( P \) and \( Q \) are the two intersections of the circles. A line passing through point \( P \) intersects the first circle at point \( A \) and the second circle at point \( B \), with \( P \) between \( A \) and \( B \). Another line passing through point \( Q \) and parallel to \( AB \) intersects the first circle at point \( D \) and the second circle at point \( C \). Find the area of the quadrilateral \( ABCD \), given that \( AP = 2 \) and \( PB = 1 \).

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