
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 the parametric equations of line $L$ as $\begin{cases} x=2+t \\ y=2-2t \end{cases}(t\text{ is a parameter})$, establish a polar coordinate system with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of curve $C$ is given by $\rho=\frac{2}{\sqrt{1+3\cos ^{2}\theta}}$.

(1) Write the polar equation of line $L$ and the rectangular equation of curve $C$.

(2) Through any point $P$ on curve $C$, draw a line $l$ such that the angle between line $l$ and line $L$ is $\frac{\pi}{3}$. Let $A$ be the intersection point of line $l$ and line $L$. Find the maximum value of $|PA|$.

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