
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
Let $F$ be the right focus of the ellipse $C:\frac{x^{2}}{2}+y^{2}=1$. A line $AB$ passing through $F$ with an acute angle intersects the ellipse $C$ at points $A$ and $B$.

(1) Assume $\overrightarrow{OE} = \overrightarrow{OA} + \overrightarrow{OB}$, where point $E$ is on the ellipse. Find the slope of line $AB$.

(2) If chord $QE$ is perpendicular to $AB$, and $\overrightarrow{EF} \parallel \overrightarrow{FQ}$, find the maximum area of the quadrilateral $AEBQ$.

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