
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 $C: x^2 - y^2 = 1$ and the line $L: y = kx - 1$,

$(1)$ If $L$ intersects $C$ at two distinct points, find the range of the real number $k$.

$(2)$ If $L$ intersects $C$ at points $A$ and $B$, and $O$ is the origin, and the area of $\triangle AOB$ is $\sqrt{2}$, find the value of the real number $k$.

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