
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 that the two foci of an ellipse are $F_1(-\sqrt{3}, 0)$ and $F_2(\sqrt{3}, 0)$, and it passes through the point $(1, \frac{\sqrt{3}}{2})$.  
(1) Find the equation of the ellipse;  
(2) Suppose a line $l$ passing through the point $A(0, 2)$ intersects the ellipse at two distinct points $M$ and $N$ (where $M$ lies between $A$ and $N$), find the range of the ratio of the areas of triangles $\triangle OAM$ and $\triangle OAN$.

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