
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 line $l$: $\begin{cases} x=1+ \frac{1}{2}t \\ y= \frac{ \sqrt{3}}{6}t\end{cases} (t$ is the parameter$)$ and the curve $C_{1}$: $\begin{cases}x=\cos \theta \\ y=\sin \theta\end{cases} (\theta$ is the parameter$)$:

(1) Suppose that $l$ intersects $C_{1}$ at points $A$ and $B$. Find $|AB|$;

(2) If the $x$-coordinates of all points on curve $C_{1}$ are compressed to $\frac{1}{2}$ of their original values, and the $y$-coordinates are compressed to $\frac{\sqrt{3}}{2}$ of their original values, a new curve $C_{2}$ is obtained. Let $P$ be a moving point on curve $C_{2}$. Find the maximum distance from point $P$ to line $l$.

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