
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
Elective 4-4: Coordinate System and Parametric Equations

Given the parametric equations of line $l$ as $\begin{cases} & x=-1-\dfrac{\sqrt{3}}{2}t, \\ & y=\sqrt{3}+\dfrac{1}{2}t \\ \end{cases}$ (where $t$ is the parameter), with the origin as the pole, the positive half-axis of $x$ as the polar axis, and the same unit length established for the polar coordinate system, the polar equation of circle $C$ is $\rho =4\sin \left( \theta -\dfrac{\pi }{6} \right)$.

$(1)$ Find the Cartesian coordinate equation of circle $C$;

$(2)$ If $P(x,y)$ is the common point of line $l$ and the circular area $\rho \leqslant {4\sin }\left( \theta -\dfrac{\pi }{6} \right)$, find the range of values for $\sqrt{3}x+y$.

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