
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, its center is at the origin, the left focus is at $F\left(-\sqrt{3},0\right)$, and it passes through the point $D\left(2,0\right)$.

(1) Find the standard equation of this ellipse;

(2) Let point $A\left(1,\frac{1}{2}\right)$, and $P$ is 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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