
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
In the plane Cartesian coordinate system $xOy$, the radius of circle $C$ is $r=2$, and the center of the circle is the intersection point $C$ of the lines $l_{1}$: $2x-y-4=0$ and $l_{2}$: $x-y-1=0$. <br/>$(1)$ Find the equation of circle $C$; <br/>$(2)$ Determine the positional relationship between the line $l_{3}$: $4x-3y-1=0$ and circle $C$. If they intersect, let the intersection points be $A$ and $B$, and find the length of the chord $|AB|$.

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