
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 parametric equation of curve $C$ is $\begin{cases} x=1+\sqrt{3}\cos \theta \\ y=\sqrt{3}\sin \theta \end{cases}$, where $\theta$ is the parameter. Taking the origin 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 line $l$ is $\rho \sin (\theta +\frac{\pi }{3})=\frac{\sqrt{3}}{2}m$.

$(1)$ When $m=3$, determine the positional relationship between line $l$ and curve $C$;

$(2)$ If there exists a point on curve $C$ such that the distance to line $l$ equals $\frac{\sqrt{3}}{2}$, find the range of the real number $m$.

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