
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 right focus of the hyperbola $C: x^2-\frac{y^2}{3}=1$ coincides with the focus of the parabola $E$.  
$(1)$ Find the standard equation of the parabola $E$.  
$(2)$ If the line $l$ passing through the right vertex of the hyperbola $C$ with a slope of $2$ intersects the parabola $E$ at points $M$ and $N$, find the length of the line segment $MN$.

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