
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 plane Cartesian coordinate system $xOy$, the graph corresponding to the curve $C: x^{2}+y^{2}=1$ undergoes a stretching transformation $\left\{\begin{array}{l}{{x'}=2x}\\{{y'}=\sqrt{3}y}\end{array}\right.$ to obtain the graph ${C'}$.  
$(1)$ Write down the Cartesian equation of curve ${C'}$;  
$(2)$ Point $P$ lies on curve ${C'}$, find the minimum distance from point $P$ to the line $l: \sqrt{3}x+y-6=0$ and the coordinates of point $P$ at that moment.

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