
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 a circle defined by the parametric equations $x = 2\cos\theta$ and $y = 2\sin\theta$, each point on the circle is transformed such that its x-coordinate remains the same, while its y-coordinate becomes half of its original value. This transformation results in a new curve $C$.

1. Find the Cartesian equation of $C$.
2. Let $l$ be the line defined by the equation $x + 2y - 2 = 0$. Denote the points of intersection between $l$ and $C$ as $P\_1$ and $P\_2$. Using the origin as the pole and the positive x-axis as the polar axis, find the polar equation of the line that passes through the midpoint of the segment $P\_1P\_2$ and is perpendicular to $l$.

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