
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 \( y^2 = 4x \) with its focus at \( F \), a line passing through the focus \( F \) and inclined at an angle \( \theta \left(0 < \theta < \frac{\pi}{2}\right) \) intersects the parabola at points \( A \) and \( B \). Points \( A_O \) and \( B_O \) (where \( O \) is the origin) intersect the directrix at points \( B' \) and \( A' \), respectively. Determine the area of the quadrilateral \( ABB'A' \).

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