
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 $l$: $\begin{cases} x=1+\frac{1}{2}t \\ y=\frac{\sqrt{3}}{2}t \end{cases} (t \text{ is a parameter})$ and the curve $C_{1}$: $\begin{cases} x=\cos\theta \\ y=\sin\theta \end{cases} (\theta \text{ is a parameter})$,
(1) Find the distance between the points $A$ and $B$ where $l$ intersects with $C_{1}$;
(2) If the horizontal coordinates of the points on the curve $C_{1}$ are stretched by a factor of $\sqrt{3}$, and the vertical coordinates are stretched by a factor of $3$, a new curve $C_{2}$ is created. Find the maximum distance from a moving point $P$ on the curve $C_{2}$ to the line $l$.

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