
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 that circle $C$ passes through two points $A(-1,1)$ and $B(3,5)$, and the center of the circle $C$ lies on the line $2x-y-5=0$.
$(1)$ Find the equation of circle $C$;
$(2)$ A line $l$ passing through point $P(1,5)$ intersects circle $C$ at points $M$ and $N$, and $|MN|=4\sqrt{3}$. Find the equation of line $l$.

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