
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 $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a > b > 0$) with eccentricity $\frac{\sqrt{3}}{2}$, and the area of the triangle formed by one endpoint of the ellipse's minor axis and its two foci is $\sqrt{3}$. The equation of circle C is $(x-a)^{2} + (y-b)^{2} = \left(\frac{a}{b}\right)^{2}$.  
(1) Find the equations of the ellipse and circle C;  
(2) A line $l$ passes through the origin $O$ and intersects circle C at points $A$ and $B$. If $\overrightarrow{CA} \cdot \overrightarrow{CB} = -2$, find the equation of line $l$.

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