
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 $C: x^{2}+(y-4)^{2}=r^{2}$, and a line $l$ passing through point $M(-2,0)$  
(I) If the radius of circle $C$ is $r=2$, and line $l$ is tangent to circle $C$, find the equation of line $l$;  
(II) If the slope angle of line $l$ is $\alpha=135^{\circ}$, and line $l$ intersects circle $C$ at points $A$ and $B$, with the chord length $|AB|=2\sqrt{2}$, find the equation of circle $C$.

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