
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 parametric equation of line $l$ as $\begin{cases} x=-1- \frac { \sqrt {3}}{2}t \\ y= \sqrt {3}+ \frac {1}{2}t\end{cases}$ (where $t$ is the parameter), with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of circle $C$ is $\rho=4\cos (\theta- \frac {2\pi}{3})$.  
$(1)$ Find the Cartesian coordinate equation of circle $C$;  
$(2)$ If $P(x,y)$ is the common point of line $l$ and the circular area $\rho\leqslant 4\cos (\theta- \frac {2\pi}{3})$, find the range of $\sqrt {3}x+y$.

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