
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 center of the ellipse $C$ is at the origin, its right focus is at $(\sqrt{3}, 0)$, and its eccentricity is $\frac{\sqrt{3}}{2}$.

(I) Find the equation of the ellipse $C$.

(II) If the line $l: y = kx + \sqrt{2}$ always intersects the ellipse $C$ at two distinct points $A$ and $B$, and $\overrightarrow{OA} \cdot \overrightarrow{OB} > 2$ (where $O$ is the origin), find the range of values for the real number $k$.

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