
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 located on the $x$-axis, the lower vertex is at $D(0,-1)$, and the eccentricity $e= \frac{ \sqrt{6}}{3}$. A line $L$ passing through the point $M(1,0)$ intersects the ellipse at points $A$ and $B$.
(I) Find the standard equation of the ellipse;
(II) Find the range of values for $|AM|$;
(III) Determine if there exists a fixed point $P$ on the $x$-axis such that $\angle MPA = \angle MPB$. If it exists, find the coordinates of point $P$; if not, explain the reason.

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