
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 a moving circle that passes through a fixed point $F(0,1)$ and is tangent to the line $y=-1$.
(1) Find the equation of the curve $C$ on which the center $M$ of the moving circle lies;
(2) A line $l$ passes through a point $P(x_{0}, y_{0})$ on the curve $C$ and is perpendicular to the tangent line of $C$ at point $P$. The other intersection point of the line $l$ and the curve $C$ is $Q$. When $x_{0} = \sqrt{2}$, find the area of $\triangle OPQ$.

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