
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 plane with its center at the origin, its left focus at $F(-\sqrt{3}, 0)$, and passing through point $D(2, 0)$:
1. Find the standard equation of the ellipse;
2. Let point $A(1, \frac{1}{2})$ and $P$ be a moving point on the ellipse. Find the trajectory equation of the midpoint $M$ of segment $PA$.

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