
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 that segment AB passes through the center of the ellipse $\frac{x^2}{8} + \frac{y^2}{4} = 1$ with points A and B on the ellipse, and $F_1$, $F_2$ being the two foci of the ellipse, find the maximum area of the quadrilateral $F_1AF_2B$.

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