
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 an ellipse with equation $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ (where $a>b>0$), the left and right foci are $F_1$ and $F_2$, respectively. Point $A$ lies on the ellipse and the eccentricity is $$\frac{\sqrt{2}}{2}$$. The line segment $AF_2$ is perpendicular to the x-axis, and $|AF_2| = \sqrt{2}$.
1. Determine the equation of the ellipse.
2. If point $A$ is in the first quadrant and a line $l$ passing through point $A$ intersects the ellipse at another point $B$, find the maximum area of triangle $\triangle AOB$.

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