
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 with center $C$ passing through points $A(1,-1)$ and $B(4,2)$, and the center $C$ lies on the line $x-y+1=0$. <br/>$(1)$ Find the standard equation of circle $C$. <br/>$(2)$ Find the equation of the tangent line passing through point $M(-2,1)$. <br/>$(3)$ Find the chord length $|MN|$ intercepted by the circle on the x-axis.

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