
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 equations of curve $C$ are $\{\begin{array}{l}x=\sqrt{3}\cos2t,\\ y=2\sin t\end{array}(t$ is the parameter). Using the origin as the pole and the positive x-axis as the polar axis, it is known that the polar equation of the line $l$ is $\rho \sin (\theta +\frac{π}{3})+m=0$.

$(1)$ Write down the Cartesian equation of $l$;

$(2)$ If $l$ has common points with $C$, find the range of values for $m$.

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