
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 that passes through the three intersection points of the parabola $y=x^2-4x+3$ and the coordinate axes.

(1) Find the equation of circle C;  
(2) Suppose the line $2x-y+2=0$ intersects circle C at points A and B, find $|AB|$.

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