
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\dfrac{2\sqrt{2}}{3}$, and a triangle formed by a point on the ellipse and the two foci of the ellipse has a perimeter of $6+4\sqrt{2}$.

(Ⅰ) Find the equation of the ellipse $C$;  
(Ⅱ) Let line $l$: $x=ky+m$ intersect ellipse $C$ at points $A$ and $B$. If the circle with diameter $AB$ passes through the right vertex $D$ of the ellipse, find the maximum area of $\triangle ABD$.

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