
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 parabolic equation \( y^2 = 2px \) with \( p > 0 \) and focus \( F \), a line passing through \( F \) and having an inclination angle \( \theta \) intersects the parabola at points \( A \) and \( B \). Find the maximum area of the triangle \( \triangle ABO \) (where \( O \) is the origin).

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