
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 $x^2-y^2=4$ and the line $l:y=k(x-1)$, discuss the range of the real number $k$ under the following conditions:

$(1)$ The line $L$ and the hyperbola have two common points.

$(2)$ The line $L$ and the hyperbola have exactly one common point.

$(3)$ The line $L$ and the hyperbola have no common points.

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