
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
Determine the standard equations of the conic sections that satisfy the following conditions:
(1) A hyperbola with an eccentricity of $\sqrt{3}$ and foci at $(-5\sqrt{3}, 0)$ and $(5\sqrt{3}, 0)$.
(2) An ellipse with an eccentricity $e = \frac{1}{2}$ and directrices $y = \pm4\sqrt{3}$.
(3) A parabola with its focus on the positive y-axis and distance to the directrix equal to 4.

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