
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 the ellipse C: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$$ with eccentricity $$\frac{1}{2}$$, and it passes through the point P(1, $$\frac{3}{2}$$).  
(1) Find the equation of the ellipse C.  
(2) Let F be the right focus of the ellipse C, and let M be a point on the ellipse, construct a circle centered at M with radius MF. What condition must point M satisfy for circle M to have two intersection points with the y-axis?  
(3) Assuming the circle M intersects the y-axis at points D and E, find the maximum distance between points D and E.

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