
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 $M: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{2 \sqrt{2}}{3}$, and a triangle formed by a point on the ellipse and the ellipse's two foci has a perimeter of $6+4 \sqrt{2}$.
(I) Find the equation of ellipse $M$;
(II) Let line $l$ intersect ellipse $M$ at points $A$ and $B$, and a circle with diameter $AB$ passes through the right vertex $C$ of the ellipse. Find the maximum area of $\triangle ABC$.

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