
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 vertex of a parabola is the center of the hyperbola $x^{2}-y^{2}=1$, and the focus of the parabola is the right vertex of the hyperbola $(1,0)$,
(1) find the standard equation of the parabola;
(2) if line $l$ passes through point $C(2,1)$ and intersects the parabola at points $M$ and $N$, does there exist a line $l$ such that $C$ is the midpoint of chord $MN$? If it exists, find the equation of line $l$; if not, explain the reason.

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