
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
The ellipse $\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1 \ (a > b > 0)$ has two foci $F_1(0, -c)$ and $F_2(0, c) \ (c > 0)$, with eccentricity $e = \frac{\sqrt{3}}{2}$, and the shortest distance from a focus to a point on the ellipse is $2 - \sqrt{3}$. Find the equation of the ellipse.

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