
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 line $l$ are given by $\begin{cases} x=1+\frac{t}{2} \\ y=2+\frac{\sqrt{3}}{2}t \end{cases} (t \text{ is the parameter}).$ Establish a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis. The polar equation of curve $C$ is $\rho=2\sqrt{3}\sin\theta$.

(I) Find the Cartesian equation of line $l$ and curve $C$.
(II) Let $A$ and $B$ be the intersection points of line $l$ and curve $C$. Find the value of $|AB|$.

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