
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 $C_{1}$: $\begin{cases}x=\cos \theta \\ y=\sin \theta\end{cases} (\theta$ is a parameter$)$:

(1) Assume that $l$ intersects $C_{1}$ at two points $A$ and $B$, find $|AB|$;

(2) If the horizontal coordinates of all points on curve $C_{1}$ are compressed to $\frac{1}{2}$ times the original, and the vertical coordinates are compressed to $\frac{ \sqrt{3}}{2}$ times the original, resulting in curve $C_{2}$. A moving point $P$ is located on curve $C_{2}$, find the minimum value of its distance to line $l$.

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