
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 the hyperbola $E: \frac{x^2}{a^2} - y^2 = 1$ ($a > 0$) has an eccentricity of $\sqrt{2}$, and the line $y = kx - 1$ intersects the right branch of the hyperbola $E$ at points $A$ and $B$.  
$(1)$ Find the range of values for $k$;  
$(2)$ If $|AB| = 6\sqrt{3}$, and point $C$ is a point on the hyperbola, and $\overrightarrow{OC} = m(\overrightarrow{OA} + \overrightarrow{OB})$, find the values of $k$ and $m$.

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