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

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