
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 major axis on the $x$-axis and an eccentricity of $e$, if the maximum distance between a point on the circle $C: x^{2}+\left(y-\frac{3}{2}\right)^{2}=1$ and a point on the ellipse is $1 + \sqrt{7}$, determine the equation of this ellipse.

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