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

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