
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 the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, the circle $x^{2}+y^{2}=a^{2}+b^{2}$ is called the "auxiliary circle" of the ellipse $C$. The minor axis of the ellipse $C$ is $2$, and the eccentricity is $\frac{\sqrt{6}}{3}$.
(I) Find the equation of the ellipse $C$;
(II) If the line $l$ intersects the ellipse $C$ at points $A$ and $B$, and its "auxiliary circle" at points $C$ and $D$, find the maximum area of $\triangle AOB$ when $|CD|= \sqrt{13}$.

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