
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 polar coordinate equation of curve $C$ is $\rho=2$, and a rectangular coordinate system is established with the pole as the origin and the positive semi-axis of the polar axis as the $x$-axis. The parametric equations of line $l$ are $\begin{cases} x=3-\frac{1}{2}t \\ y=2+\frac{\sqrt{3}}{2}t \end{cases}$ ($t$ is the parameter).

(I) Write out the equations of line $l$ and curve $C$ in the rectangular coordinate system;

(II) Suppose curve $C$ undergoes a stretch transformation $\begin{cases} x'=x \\ y'=2y \end{cases}$ to obtain curve $C'$. For any point $M(x_{0},y_{0})$ on curve $C'$, find the range of values for $2\sqrt{3}{x_{0}}+{{y_{0}}}$.

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