
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 and its right focus $F(2,0)$, and the distance from $F$ to one of the asymptotes of the hyperbola is $1$.
$(I)$ Find the equation of the hyperbola $C$;
$(II)$ If the line $l: y=kx+2$ 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 $k$.

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