
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 equation that fits the following conditions:
1. For the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, the top vertex is $(0,2)$, and the eccentricity is $\frac{\sqrt{3}}{2}$. Find the equation of the ellipse $C$.
2. The standard equation of a hyperbola with foci on the $y$-axis, a focal distance of $16$, and an eccentricity $e = \frac{4}{3}$.

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