
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
On curve $C_{1}$, any point $M$ satisfies $|MF_{1}|+|MF_{2}|=4$, where $F_{1}(-\sqrt{3},0)$, $F_{2}(\sqrt{3},0)$. The focus of parabola $C_{2}$ is the intersection point of the line $y=x-1$ and the $x$-axis, and its vertex is the origin $O$.
$(1)$ Find the standard equations of $C_{1}$ and $C_{2}$;
$(2)$ Is there a line $l$ that satisfies the conditions: $(i)$ passes through the focus $F$ of $C_{2}$; $(ii)$ intersects $C_{1}$ at two different points $M$, $N$, and satisfies $\overrightarrow{OM} \perp \overrightarrow{ON}$? If it exists, find the equation of line $l$; if not, explain why.

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