
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 a plane Cartesian coordinate system, consider the curve C<sub>1</sub> such that for each point on C<sub>1</sub>, the x-coordinate remains unchanged but the y-coordinate is halved to obtain the curve C<sub>2</sub>. Using the origin O as the pole and the positive x-axis as the polar axis, the polar equation of the curve C<sub>1</sub> is given by ρ = 2.
(Ⅰ) Determine the parametric equation for the curve C<sub>2</sub>;
(Ⅱ) Two lines l<sub>1</sub> and l<sub>2</sub> pass through the origin O and intersect the curve C<sub>2</sub> at points A and C, and B and D respectively, such that they are symmetric with respect to the y-axis, and point A lies in the first quadrant. When the perimeter of the quadrilateral ABCD is maximized, find the equation of the line l<sub>1</sub> in the standard form.

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