
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
The right focus of the ellipse $\frac{{x^2}}{{a^2}}+\frac{{y^2}}{{b^2}}=1\left(a \gt b \gt 0\right)$ is $F$, the right vertex is $A$, and the top vertex is $B$, satisfying $\frac{{|BF|}}{{|AB|}}=\frac{{\sqrt{3}}}{2}$. <br/>$(1)$ Find the eccentricity $e$ of the ellipse; <br/>$(2)$ The line $l$ intersects the ellipse at a unique point $M$ and intersects the $y$-axis at $N$ ($N$ is different from $M$). Let $O$ be the origin. If $|OM|=|ON|$ and the area of $\triangle OMN$ is $\sqrt{3}$, find the standard equation of the ellipse.

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