
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, with the vertex at the origin $O$.
(1) Find the standard equations of $C_1$ and $C_2$;
(2) Is there a line $l$ that satisfies the conditions: ① passes through the focus $F$ of $C_2$; ② 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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