
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 the point F(0,1) is the focus of the parabola $x^2=2py$,

(1) Find the equation of the parabola C;
(2) Points A, B, and C are three points on the parabola such that $\overrightarrow{FA} + \overrightarrow{FB} + \overrightarrow{FC} = \overrightarrow{0}$, find the maximum value of the area of triangle ABC.

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