
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
Selected Exercise $(4-4)$: Parametric Equations and Polar Coordinates

In the Cartesian coordinate system $xoy$, the parametric equations of line $l$ are given by $\begin{cases} & x=\frac{1}{2}t \\ & y=m+\frac{\sqrt{3}}{2}t \end{cases} (t \text{ is the parameter})$, and in the polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, the polar equation of curve $C$ is given by ${\rho }^{2}-2\rho \cos \theta -4=0$.

$(1)$ If line $l$ and curve $C$ have no common points, find the range of values for $m$.

$(2)$ If $m=0$, find the length of the chord cut off by line $l$ on curve $C$.

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