
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 a line $l$ intersects the ellipse $C: \frac{x^{2}}{2}+y^{2}=1$ at points $A$ and $B$, and $O$ is the origin.
(1) Find the maximum area of triangle $\triangle A O B$ and the equation of the line $l$ at that time.
(2) Let the set $L=\{l \mid l \text{ makes the area of } \triangle A O B \text{ attain its maximum} \}$, and let $l_{1}, l_{2}, l_{3}, l_{4} \in L$ such that $l_{1} || l_{2}$, $l_{3} || l_{4}$, with $k_{1}, k_{2}, k_{3}, k_{4}$ being the slopes of $l_{1}, l_{2}, l_{3}, l_{4}$ respectively and satisfying $k_{1}+k_{2}+k_{3}+k_{4}=0$. Find the minimum area of the quadrilateral formed by these four lines.

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