
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 $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a > b > 0)$ where the length of the major axis is twice the length of the minor axis.<br/>$(1)$ Find the eccentricity $e$ of the ellipse;<br/>$(2)$ A line $l$ passes through the point $N(0,2)$ and intersects the ellipse at a unique point $M$. $O$ is the origin. Find the equation of the ellipse when the area of $\triangle OMN$ is maximized.

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