
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
Find the standard equation of the ellipse that satisfies the following conditions:<br/>$(1)$ The coordinates of the two foci are $\left(0,-2\right)$ and $\left(0,2\right)$, and the ellipse passes through the point $({-\frac{3}{2},\frac{5}{2}})$;<br/>$(2)$ The ellipse passes through the points $({2,-\sqrt{2}})$ and $({-1,\frac{{\sqrt{14}}}{2}})$.

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