
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 with its foci on the x-axis and its lower vertex at D(0, -1), the eccentricity of the ellipse is $e = \frac{\sqrt{6}}{3}$. A line L passes through the point P(0, 2).
(Ⅰ) Find the standard equation of the ellipse.
(Ⅱ) If line L is tangent to the ellipse, find the equation of line L.
(Ⅲ) If line L intersects the ellipse at two distinct points M and N, find the maximum area of triangle DMN.

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