
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 an ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity $e = \frac{1}{2}$, a line passing through the focus and perpendicular to the x-axis intersects the ellipse, forming a line segment with a length of 3.

1. Find the equation of the ellipse.
2. Suppose P is a fixed point in the Cartesian plane, and a moving line $l: y = \frac{1}{2}x + t$ intersects the ellipse at points A and B. When the slopes of both PA and PB exist and the sum of their slopes is a constant unrelated to $t$, find the coordinates of all points P that satisfy these conditions.

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