
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 line $l$ are $\left\{{\begin{array}{l}{x=\sqrt{3}+\frac{1}{2}t}\\{y=2+\frac{{\sqrt{3}}}{2}t}\end{array}\right.}\right.$ (where $t$ is the parameter) and the parametric equations of curve $C$ are $\left\{{\begin{array}{l}{x=4\cos\theta}\\{y=4\sin\theta}\end{array}\right.}\right.$ (where $\theta$ is the parameter). Find:<br/>$(1)$ The Cartesian equation of curve $C$;<br/>$(2)$ If line $l$ intersects curve $C$ at points $A$ and $B$, find the length of segment $AB$.

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