
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 the line $l$ are $\left\{\begin{array}{l}{x=\frac{1}{2}t}\\{y=\frac{\sqrt{3}}{2}t}\end{array}\right.$ (where $t$ is a parameter). In the polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis, the polar equation of the curve $C$ is ${\rho^2}{\cos^2}\theta+5\sqrt{3}\rho\cos\theta-\rho\sin\theta+3=0$.  
$(1)$ Find the polar equation of the line $l$ and the Cartesian equation of the curve $C$;  
$(2)$ Find the length of the line segment intercepted by the line $l$ and the curve $C$.

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