
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 $C\_1$: $\begin{cases} x=\cos θ \\ y=\sin θ \end{cases}(θ\text{ is a parameter})$

(1) Find the distance $|AB|$ between the points $A$ and $B$ where $l$ and $C\_1$ intersect;

(2) If the horizontal coordinates of each point on the curve $C\_1$ are compressed to $\frac{1}{2}$ of the original, and the vertical coordinates are compressed to $\frac{ \sqrt{3}}{2}$ of the original, a new curve $C\_2$ is obtained. Find the minimum value $d$ of the 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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