
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 the ellipse $\_(x)\_^2 + (\_*(m)*\+3)\_*(y)\_^2 = \_*(m)*\*(m > 0)$ with an eccentricity $e= \frac{ \sqrt{3}}{2} $, find the value of $m$, the length of the major and minor axes, the coordinates of the foci, and the coordinates of the vertices.

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