
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 polar equation of circle $C$ is $\rho=2$, and a Cartesian coordinate system is established with the pole as the origin and the positive half-axis of the polar axis as the $x$-axis. If the line $l$: $kx+y+3=0$ is tangent to circle $C$.  
Find $(1)$ the Cartesian equation of circle $C$;  
$(2)$ the value of the real number $k$.

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