
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 line $l: \rho\sin(\theta - \frac{\pi}{4}) = 4$ and the circle $C: \rho = 2k\cdot\cos(\theta + \frac{\pi}{4})$ ($k \neq 0$), if the minimum distance from a point on line $l$ to a point on circle $C$ equals 2.  
(1) Find the Cartesian coordinates of the center of circle $C$;  
(2) Find the value of $k$.

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