
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 $D(2,0)$. Let point $A(1,\frac{1}{2})$.

(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 the line segment $PA$;
(3) A line passing through the origin $O$ intersects the ellipse at points $B$ and $C$. Find the maximum area of $\triangle ABC$.

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