
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 equations of line $l$ as $\begin{cases} x = -1 - \frac{\sqrt{3}t}{2} \\ y = \sqrt{3} + \frac{1}{2}t \end{cases}$ $(t$ is the parameter$)$, establish a polar coordinate system with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axis. The polar coordinate equation of circle $C$ is $\rho = 4 \sin(\theta - \frac{\pi}{6})$.

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

(2) Point $P(x, y)$ is the intersection point of line $l$ and the circular region $\rho \leqslant 4 \sin(\theta - \frac{\pi}{6})$. Find the range of values for $\sqrt{3}x + y$.

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