
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
A line and a circle intersect at exactly one point, while another line intersects a hyperbola given by the equation \(x^2 - y^2 = 1\), but neither line is tangent to the hyperbola. If the first line also intersects the hyperbola, how many total points of intersection might there be between the lines and the hyperbola.

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