
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 a circle C with its center on the x-axis that passes through points (0, 0) and (-1, 1), and the equation of circle D is $(x-4)^2+y^2=4$  
(1) Find the equation of circle C;  
(2) A moving point P on circle D draws two tangents to circle C, intersecting the y-axis at points A and B, respectively. Find the range of $|AB|$.

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