
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 that the center of circle C lies on the negative half of the y-axis, with a radius of 3, and that the line $y+1=0$ is tangent to circle C,   
(1) find the equation of circle C;  
(2) if circle C intersects the line $x-y-1=0$ at points A and B, find the distance $|AB|$ between points A and B.

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