
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 $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ with an imaginary axis length of $2$ and a focal length of $2\sqrt{3}$, find the equations of the asymptotes and the equation of the ellipse with the hyperbola's foci as its vertices and the hyperbola's vertices as its foci.

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