
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 $x^2=4y$ with focus $F$, and $P$ is a moving point on the parabola.  
(1) When $|PF|=2$, find the coordinates of point $P$;  
(2) A line passing through $F$ with a slope of 1 intersects the parabola at two points $A$ and $B$. If $P$ is on the arc $AB$, find the maximum value of the area of $\triangle PAB$.

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