
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 circle $C$ with a radius of $4$ that is tangent to the line $l_{1}: 3x-4y+8=0$, and the center of the circle $C$ lies on the negative y-axis.  
$(1)$ Find the equation of the circle $C$;  
$(2)$ Given that the line $l_{2}: kx-y+3=0$ intersects the circle $C$ at points $A$ and $B$, and the area of $\triangle ABC$ is $8$, find the equation of the line $l_{2}$.

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