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

1. The line $l$ intersects the curve $C_{1}$ at points $A$ and $B$. Calculate the distance $|AB|$.
2. If the horizontal coordinates of each point on the curve $C_{1}$ are stretched to $\sqrt{3}$ times the original, and the vertical coordinates are stretched to 3 times the original, a new curve $C_{2}$ is obtained. For a moving point $P$ on the curve $C_{2}$, find the maximum distance from $P$ to the line $l$.

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