
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 center of the ellipse is at the origin, its foci are on the $x$-axis, the focal distance is $2$, and it passes through point $A(-1, \frac{3}{2})$;  
$(1)$ Find the equation of the ellipse that meets these conditions;  
$(2)$ Find the coordinates of the vertices of the ellipse, the length of the major axis, the length of the minor axis, and the eccentricity.

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