
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 parabola \( y^{2}=2px \) where \( p>0 \), its focus is \( F \). A line passing through \( F \) and having an inclination angle \(\theta\) intersects the parabola at points \( A \) and \( B \). The line connecting point \( A \) and the vertex \( O \) of the parabola intersects the directrix at point \( B' \). The line connecting point \( B \) and \( O \) intersects the directrix at point \( A' \). Find the area of the quadrilateral \( ABB'A' \).

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