
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: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a > b > 0$) with an eccentricity of $\frac{2\sqrt{2}}{3}$, and a triangle formed by a point on the ellipse and its two foci has a perimeter of $6 + 4\sqrt{2}$.
(Ⅰ) Find the equation of ellipse C;
(Ⅱ) Suppose the line $l: x = ky + m$ intersects ellipse C at points A and B. If the circle with AB as its diameter 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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