
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
Let the left focus of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ be $F$, and the top vertex be $B$. Given that the eccentricity of the ellipse is $\dfrac{\sqrt{5}}{3}$, the coordinates of point $A$ are $(b,0)$, and $|FB| \cdot |AB| = 6\sqrt{2}$.

(Ⅰ) Find the equation of the ellipse;

(Ⅱ) Let the line $l: y = kx (k > 0)$ intersect the ellipse in the first quadrant at point $P$, and intersect line $AB$ at point $Q$. If $\dfrac{|AQ|}{|PQ|} = \dfrac{5\sqrt{2}}{4}\sin \angle AOQ$ (where $O$ is the origin), find the value of $k$.

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