
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 an ellipse \( C: \frac{x^{2}}{2}+y^{2}=1 \) at points \( A \) and \( B \), and \( O \) is the origin of the coordinates.
1. Find the maximum area of triangle \(\triangle AOB\) and the equation of the line \( l \) when this maximum area is achieved.
2. Let \( L=\{l \mid \) the line \( l \) gives the maximum area of \(\triangle AOB \) \}. If \( l_{1}, l_{2}, l_{3}, l_{4} \in L \) such that \( l_{1} \parallel l_{2}, l_{3} \parallel l_{4} \), and \( k_{1}, k_{2}, k_{3}, k_{4} \) are the slopes of the corresponding lines such that \( k_{1} + k_{2} + k_{3} + k_{4} = 0 \), 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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