
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 in the Cartesian coordinate system xOy, the parametric equation of the line l is
$$
\begin{cases}
x = \sqrt{2} \cdot t \sin \frac{\pi}{6} \\
y = t \cos \frac{7\pi}{4} - 6 \sqrt{2}
\end{cases}
$$
where t is the parameter. By setting the origin O as the pole and Ox as the initial line of the polar coordinate system, the polar equation of the circle C is
$$
\rho = 4\cos\left(\theta + \frac{\pi}{4}\right).
$$
(1) Find the standard equation of line l and the Cartesian coordinates of the center C of the circle;
(2) Find the minimum distance from a point on circle C to line l.

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