
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
A circle $C$ with a radius of $3$ passes through point $A(1,-1)$, the center of the circle $C$ lies on the line $y=2x$, and the center of the circle lies in the first quadrant.  
$(1)$ Find the equation of circle $C$;  
$(2)$ A tangent line to circle $C$ passes through point $(4,3)$, find the equation of the tangent line.

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