
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 $l$: $\rho\sin\left(\theta-\frac{\pi}{4}\right)=4$ and the circle $C$: $\rho=2k\cos\left(\theta+\frac{\pi}{4}\right)$ ($k\neq 0$), if the minimum distance from a point on line $l$ to a point on circle $C$ is equal to $2$, find the value of the real number $k$ and the rectangular coordinates of the center of circle $C$.

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