
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 an ellipse with its axes of symmetry being the coordinate axes, and one endpoint of the minor axis and the two foci form the vertices of an equilateral triangle. The shortest distance from a focus to a point on the ellipse is $\sqrt{3}$. Find the equation of this ellipse and its eccentricity.

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