
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 parabola M: $y^2 = 4x$, and the circle N: $(x-1)^2 + y^2 = r^2$ (where r is a constant, $r > 0$). The line l passing through the point (1, 0) intersects the circle N at points C and D, and intersects the parabola M at points A and B, satisfying $|AC| = |BD|$. Determine the range of possible values for the constant r such that there are exactly three lines l.

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