
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
Selected Exercise $(4-4)$: Coordinate Systems and Parametric Equations

Given that in the rectangular coordinate system $xOy$, the equation of the ellipse $(C)$ is $\frac{{y}^{2}}{16}+\frac{{x}^{2}}{4}=1$. Establish a polar coordinate system with $O$ as the pole and the non-negative semiaxis as the polar axis, using the same unit of length. The polar equation of the line is $\rho \sin \left( \theta +\frac{\pi }{3} \right)=3$.

(1) Find the rectangular equation of the line and the parametric equation of the ellipse $(C)$;

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

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