
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 $M$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\dfrac{\sqrt{6}}{3}$ and focal distance $2\sqrt{2}$. A line $l$ with slope $k$ intersects the ellipse $M$ at two distinct points $A$ and $B$.

(Ⅰ) Find the equation of the ellipse $M$;

(Ⅱ) If $k=1$, find the maximum value of $|AB|$;

(Ⅲ) Let $P(-2,0)$, the line $PA$ intersects the ellipse $M$ at another point $C$, and the line $PB$ intersects the ellipse $M$ at another point $D$. If $C$, $D$, and the point $Q(-\dfrac{7}{4}, \dfrac{1}{4})$ are collinear, find $k$.

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