
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 a hyperbola $C$ with the same foci as the ellipse $\frac{x^2}{8} + \frac{y^2}{4} = 1$ and a real semi-axis length of $\sqrt{3}$,  
(1) Find the equation of the hyperbola $C$;  
(2) If a straight line $l: y = kx + \sqrt{2}$ has two distinct intersection points $A$ and $B$ with hyperbola $C$, and $\overrightarrow{OA} \cdot \overrightarrow{OB} > 2$ (where $O$ is the origin), find the range of $k$.

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