
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 $l$ is 
$$
\begin{cases} 
x= \sqrt {3}- \frac { \sqrt {3}}{2}t \\
y=1+ \frac {1}{2}t
\end{cases}
(t \text{ is the parameter}).
$$
Taking the origin of the Cartesian coordinate system as the pole and the non-negative half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of curve $C$ is $\rho=4\cos (\theta- \frac {\pi}{6})$.  
$(1)$ Find the Cartesian coordinate equation of curve $C$;  
$(2)$ If $P(x,y)$ is the common point of line $l$ and the curve surface $\rho\leqslant 4\cos (\theta- \frac {\pi}{6})$, find the range of $\mu= \sqrt {3}x+y$.

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