
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 function $y=x^2-4x+3$ intersects the x-axis at points M and N, and intersects the y-axis at point P. A circle with center C exactly passes through points M, N, and P.
(1) Find the equation of circle C;
(2) If circle C intersects the line $x-y+n=0$ at points A and B, and the length of segment $|AB|=4$, find the value of $n$.

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