
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
In the Cartesian coordinate system $xoy$, given three points $A(-1, 0)$, $B(1, 0)$, and $C(-1, \frac{3}{2})$, there is an ellipse with foci at points $A$ and $B$ that passes through point $C$.
(I) Find the equation of the ellipse;
(II) Let point $D$ be $(0, 1)$, determine if there exists a line $l$, not parallel to the x-axis, that intersects the ellipse at two distinct points $M$ and $N$, such that $(\overrightarrow{DM} + \overrightarrow{DN}) \cdot \overrightarrow{MN} = 0$. If it exists, find the range of possible slopes for line $l$; if not, explain why;
(III) If for a point $P$ on the y-axis $P(0, n)$ (where $n \neq 0$), there exists a line $l$, not parallel to the x-axis, that intersects the ellipse at two distinct points $M$ and $N$, such that $(\overrightarrow{PM} + \overrightarrow{PN}) \cdot \overrightarrow{MN} = 0$, determine the range of values for $n$.

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