
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 the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) has an imaginary axis length of 2 and a focal distance of $2\sqrt{3}$. Find the equation of the asymptotes of the hyperbola and the equation of the ellipse whose vertices and foci are the foci and vertices of the hyperbola, respectively.

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