
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 in the Cartesian coordinate system xOy, its center is at the origin, the left focus is $F(-\sqrt{3}, 0)$, and the right vertex is D(2, 0).

1. Find the standard equation of the ellipse.
2. (Liberal Arts) If P is a moving point on the ellipse, draw a line perpendicular to the x-axis through P, and let M be the foot of the perpendicular. Extend MP to N such that P is the midpoint of MN. Find the trajectory equation of point N.
3. (Science) Given point $A(1, \frac{1}{2})$, and P is a moving point on the ellipse. Find the trajectory equation of the midpoint M of line segment PA.

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