
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 Cartesian coordinate system $xOy$, the angle of inclination of a line $l$ is $\alpha$. Its parametric equation is $\begin{cases}
x = -2 + t\cos \alpha \\
y = t\sin \alpha
\end{cases}$ ($t$ is the parameter). In the polar coordinate system with the origin $O$ as the pole and the non-negative half of the $x$-axis as the initial ray (using the same units of length), the polar equation of curve $C$ is $\rho - 4\cos \theta = 0$.

$(I)$ If line $l$ and curve $C$ have common points, find the range of values for the inclination angle $\alpha$ of line $l$.

$(II)$ Let $M(x,y)$ be any point on curve $C$. Find the range of values for $x+ \sqrt{3}y$.

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