
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 equation of ellipse $C$ is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $(a>b>0)$, and the two asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are $l_1, l_2$ with $l_1$ making an angle of 30° with the x-axis, and the focal distance of the hyperbola is $4\sqrt{2}$.
(1) Find the equation of the ellipse $C$.
(2) A line $l$ passes through the right focus $F$ of the ellipse $C$ and intersects the ellipse at points $A$ and $B$, and intersects the circle $O: x^2+y^2=a^2$ at points $D$ and $E$. When the area of $\triangle OAB$ is maximal, find the length of the chord $DE$.

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