
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 hyperbola \( C_{1}: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{2a^{2}} = 1 \) (where \( a > 0 \)), and the parabola \( C_{2} \), with its vertex at the origin \( O \). The focus of \( C_{1} \) is the same as the left focus \( F_{1} \) of \( C_{2} \). Is there a chord \( AB \) passing through the focus \( F_{1} \) of \( C_{2} \) such that the area \( S_{\triangle AOB} \) of the triangle \( AOB \) is maximized or minimized? If such a chord exists, find the equation of the line \( AB \) and the maximum or minimum value. If not, provide a reason.

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