
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 $(xOy)$, its center is at the origin, the left focus is at $F\_1(-\sqrt{3}, 0)$, and the right vertex is at $D(2, 0)$. Let point $A$ have coordinates $\left(\begin{matrix}1, \frac{1}{2} \end{matrix}\right)$.

(1) Find the standard equation of the ellipse;

(2) If $P$ 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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