
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
(1) Ellipse $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ passes through point $A(0,-1)$, and its eccentricity is $\frac{\sqrt{2}}{2}$, find the equation of ellipse $E$;
(2) Find the standard equation of the ellipse passing through points $M(2, \sqrt{2})$, $N(\sqrt{6}, 1)$.

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