
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 $l$: $\begin{cases} x=2+ \frac{1}{2}t \\ y=\frac{ \sqrt{3}}{2}t \end{cases}(t\text{ is a parameter})$, and the curve $C_{1}$: $\begin{cases} x=2\cos \theta \\ y=2\sin \theta \end{cases}(\theta\text{ is a parameter})$
(1) The line $l$ intersects the curve $C_{1}$ at points $A$ and $B$. Find the length $|AB|$;
(2) If the horizontal coordinates of all points on the curve $C_{1}$ are compressed to $\frac{1}{2}$ of their original values, and the vertical coordinates are compressed to $\frac{\sqrt{3}}{2}$ of their original values, a new curve $C_{2}$ is obtained. Let point $P$ be a moving point on the 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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