
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 hyperbola \( C_{1}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{2 a^{2}}=1(a>0) \), and a parabola \( C_{2} \) with its vertex at the origin \( O \). The focus of \( C_{1} \) is also the left focus \( F_{1} \) of \( C_{2} \). Determine if there exists a chord \( A B \) of \( C_{2} \) passing through \( F_{1} \) such that the area \( S_{\triangle A O B} \) is maximized or minimized. If it exists, find the equation of the line containing \( A B \) and the extreme value. If it doesn't exist, provide an explanation.

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