
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 $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively, and $|F\_1F\_2|=6$. The line $y=kx$ intersects the ellipse at points $A$ and $B$.
(I) If the perimeter of $\Delta AF\_1F\_2$ is $16$, find the standard equation of the ellipse.
(II) If $k=\frac{\sqrt{2}}{4}$ and points $A$, $B$, $F\_1$, and $F\_2$ are concyclic, find the eccentricity $e$ of the ellipse.
(III) Under the conditions of (II), let $P(x\_0,y\_0)$ be a point on the ellipse, and the slope of line $PA$ is $k\_1 \in (-2,-1)$. Find the range of the slope $k\_2$ of line $PB$.

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