
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
On the circle $x^2+y^2=3$, an arbitrary moving point P is chosen. A perpendicular line PD is drawn from P to the x-axis, with D being the foot of the perpendicular. It is given that $\overrightarrow{PD} = \sqrt{3} \overrightarrow{MD}$. The trajectory of the moving point M is curve C.  
(1) Find the equation of C and its eccentricity;  
(2) If a line l intersects curve C at points A and B, and the distance from the origin to line l is $\frac{\sqrt{3}}{2}$, find the maximum value of the area of $\triangle AOB$.

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