
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 an ellipse $C$ in the Cartesian coordinate system $xOy$ with the equation $\frac{y^2}{16}+\frac{x^2}{4}=1$, take point $O$ as the pole and the non-negative half of the $x$-axis as the polar axis to establish a polar coordinate system. The polar equation of line $l$ is $\rho\sin(\theta+\frac{\pi}{3})=3$.

$(1)$ Find the Cartesian equation of line $l$ and the parametric equations of the ellipse $C$.

$(2)$ Let $M(x,y)$ be any point on ellipse $C$. Find the maximum value of $|2\sqrt{3}x+y-1|$.

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