
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 the center $C$ of circle $C$ lies on the line $y=2x$, and passes through points $A(-1,0)$ and $B(3,0)$.<br/>$(1)$ Find the equation of circle $C$;<br/>$(2)$ The line $l: mx+y-3m-1=0$ intersects circle $C$ at points $E$ and $F$, and $|EF|=2\sqrt{3}$. Find the value of the real number $m$.

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