
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 \( C_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \), there is a line \( l \) that passes through the origin and intersects the ellipse at points \( A \) and \( B \).

(1) Find the maximum value of the area of triangle \( OAB \).

(2) Does there exist an ellipse \( C_{2} \) such that for every tangent line to \( C_{2} \), it intersects ellipse \( C_{1} \) at points \( A \) and \( B \), and the area \( S_{\triangle OAB} \) is maximized? If it exists, find the ellipse; if not, explain why.

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