
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 that line $l$ passes through the focus of the parabola $(C)$, and is perpendicular to $(C)$'s axis of symmetry. The line $l$ intersects with $(C)$ at points $A$ and $B$, with $|AB|=12$. $P$ is a point on the directrix of $(C)$. Determine the area of $\triangle ABP$.

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