
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 that the equation of ellipse $C$ is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, the asymptotes of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ are $l_{1}$ and $l_{2}$, and the angle between $l_{1}$ and the $x$-axis is $30^{\circ}$, and the focal length of the hyperbola is $4 \sqrt{2}$.
(1) Find the equation of ellipse $C$;
(2) A line $l$ passing through the right focus $F$ of ellipse $C$ intersects ellipse $C$ at points $A$ and $B$, and circle $O$: $x^{2} + y^{2} = a^{2}$ at points $D$ and $E$. When the area of triangle $OAB$ is maximum, find the length of chord $DE$.

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