
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 line $4x+3y-12=0$ intersects a circle $C$ with center at point $C(1,1)$, and the length of the chord is $2\sqrt{3}$.  
$(1)$ Find the equation of circle $C$;  

$(2)$ Find the equation of the tangent line to circle $C$ passing through point $(-1,2)$.

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