
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 an ellipse $C$ whose foci coincide with the vertices of the hyperbola $\frac{y^2}{3} - x^2 = 1$, and the major axis of the ellipse $C$ is $4$.
$(1)$ Find the length of the real and imaginary axes of the hyperbola and the equation of its asymptotes.
$(2)$ Find the standard equation of the ellipse $C$;
$(3)$ If it is known that the line $y = x + m$, for what value of $m$ does the line have common points with the ellipse $C$?

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