
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 equation of line $l$ as $y=-x-2$, point $P$ is the point on the parabola $C: x^2=4y$ that is closest to line $l$.
(I) Find the coordinates of point $P$;
(II) If line $m$ intersects parabola $C$ at points $A$ and $B$, and the centroid of $\triangle ABP$ is exactly the focus $F$ of parabola $C$. Find the area of $\triangle ABP$.

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