
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 hyperbola $C$ and the ellipse $\frac{x^2}{8} + \frac{y^2}{4} = 1$ have the same foci, and the real semi-axis length of the hyperbola is $\sqrt{3}$.

$(1)$ Find the equation of the hyperbola $C$;  
$(2)$ If the line $l: y = kx + \sqrt{2}$ intersects the hyperbola $C$ 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 $k$.

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