
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 a parameter), and the curve $C_1$: $\begin{cases} x=\cos \theta \\ y=\sin \theta \\ \end{cases}$ ($\theta$ is a parameter).

(1) Find the distance $|AB|$ between the points $A$ and $B$ where $l$ intersects $C_1$;
(2) If the $x$-coordinate of each point on the curve $C_1$ is compressed to $\frac{1}{2}$ of its original value, and the $y$-coordinate is compressed to $\frac{\sqrt{3}}{2}$ of its original value, a new curve $C_2$ is obtained. Let $P$ be a moving point on the curve $C_2$, find the maximum value of its distance to the line $l$.

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