
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 $\left\{{\begin{array}{l}{x=2\cos t,}\\{y=2\sin t}\end{array}}\right.$ (where $t$ is the parameter). Taking the coordinate origin as the pole and the positive x-axis as the polar axis, establish a polar coordinate system. The polar equation of the line $l$ is $ρ\cos({θ-\frac{π}{3}})+m=0$ (where $m∈{R}$).  
$(1)$ Write down the Cartesian equation of the line $l$.  
$(2)$ If the line $l$ has common points with the curve $C$, find the range of real number $m$.

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