
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 \( p > 0 \), with the focus at \( F \), a line passing through the focus \( F \) and having an angle of inclination \( \theta \) (where \( 0 < \theta < \pi \)) intersects the parabola at points \( A \) and \( B \). Line segment \( AO \) (where \( O \) is the origin) intersects the directrix at point \( B' \). Line segment \( BO \) intersects the directrix at point \( A' \). Find the area of quadrilateral \( ABB'A' \).

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