
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 the two foci of the ellipse $F_{1}(-\sqrt{3},0), F_{2}(\sqrt{3},0)$, and it passes through the point $(1, \frac{\sqrt{3}}{2})$.
$(1)$ Find the equation of the ellipse;
$(2)$ If a line $l$ passing through point $A(0,2)$ intersects the ellipse at two distinct points $M$ and $N$ (with $M$ between $A$ and $N$), find the range of the ratio of the areas of $\triangle OAM$ and $\triangle OAN$.

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