
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 curve $C$: $\frac{x^2}{4} + \frac{y^2}{9} = 1$, and the line $l$: $\begin{cases}x=2+t \\ y=2-2t\end{cases}$ (where $t$ is a parameter).

$(1)$ Write the parametric equation of curve $C$ and the standard equation of line $l$;

$(2)$ For any point $P$ on curve $C$, draw a line that forms a $30^{\circ}$ angle with line $l$ and intersects line $l$ at point $A$. Find the maximum and minimum values of $|PA|$.

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