
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
Find the standard equations that fit the following conditions:

(1) The focus is on the $x$-axis, has the same eccentricity as the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$, and passes through the point $(2, -\sqrt{3})$;

(2) The focus is on the $x$-axis, the distance between the vertices is $6$, and the asymptotes are $y = \pm \frac{1}{3}x$ of the hyperbola.

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