
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 the x-coordinate of the midpoint of $AB$ is $\sqrt{2}$, find the length of segment $AB$.

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