
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 $$E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$$ whose major axis is twice the length of its minor axis, and it passes through point C(2, 1). The point symmetric to C with respect to the origin O is point D.

(I) Find the equation of ellipse E;

(II) Point P is on ellipse E, and the slopes of lines CP and DP both exist and are not zero. Determine whether the product of the slopes of CP and DP is a constant value. If so, find this constant value; if not, explain why;

(III) A line l parallel to CD intersects ellipse E at points M and N. Find the maximum area of triangle CMN and the equation of line l at this condition.

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