
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
In the rectangular coordinate system, the horizontal coordinate of each point on the curve ($C\_1$) remains unchanged, and the vertical coordinate is shortened to half of the original, resulting in the curve ($C\_2$). With the coordinate origin ($O$) as the pole and the positive semi-axis of the $x$-axis as the polar axis, establish a polar coordinate system. The polar coordinate equation of ($C\_1$) is $p=2$.

(I) Find the parametric equation of the curve ($C\_2$);

(II) Two lines ($l\_1$) and ($l\_2$) that pass through the origin ($O$) and are symmetric about the $y$-axis intersect the curve ($C\_2$) at points $A$, $C$ and $B$, $D$, respectively, with point $A$ located in the first quadrant. When the perimeter of the quadrilateral ($ABCD$) is at its maximum, find the general equation of the line ($l\_1$).

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