
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 its center at the origin, its right focus is at $(2,0)$, and its right vertex is at $(\sqrt{3},0)$.

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