
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
In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. It is known that the parametric equation of line $l$ is $\begin{cases} x= \frac { \sqrt {2}}{2}t \\ y= \frac { \sqrt {2}}{2}t+4 \sqrt {2}\end{cases}$ (where $t$ is the parameter), and the polar equation of circle $C$ is $\rho=2\cos (\theta+ \frac {\pi}{4})$.
$(1)$ Find the Cartesian coordinates of the center of circle $C$;
$(2)$ Draw a tangent from a point on line $l$ to circle $C$ and find the minimum length of the tangent.

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