
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 the circle $C: x^{2}+y^{2}-8y+12=0$, the line $l$ passes through the point $P(-2,0)$.<br/>$(1)$ If the line $l$ is tangent to the circle $C$, find the equation of the line $l$.<br/>$(2)$ If the line $l$ intersects the circle $C$ and cuts off a chord $AB$ with $|{AB}|=2\sqrt{2}$, find the equation of the line $l$.

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