
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 centered at the origin with its foci on the $x$-axis and an eccentricity of $e=\frac{\sqrt{3}}{2}$. It is known that the maximum distance from point $P(0,\frac{3}{2})$ to any point on the ellipse is $\sqrt{7}$. Find the equation of the ellipse.

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