
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
Given that the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=t\cos\alpha\\ y=t\sin\alpha\end{array}\right.$ (where $t$ is the parameter), with the origin $O$ as the pole and the positive x-axis as the polar axis in a polar coordinate system, and the polar equation of the curve $C$ is $\rho ^{2}-2m\rho \cos \theta -4=0$ (where $m \gt 0$). <br/>$(1)$ Find the Cartesian equation of the line $l$ and the curve $C$ intersecting. <br/>$(2)$ If $m=2$, find the range of values for the chord length intercepted by the line and the curve $C$ as $\alpha$ varies.

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