
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 $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$. Point $A$ is on the ellipse with an eccentricity of $\frac{\sqrt{2}}{2}$. $AF\_2$ is perpendicular to the $x$-axis, and $|AF\_2| = \sqrt{2}$.
(1) Find the equation of the ellipse;
(2) If point $A$ is in the first quadrant, draw a line $l$ passing through point $A$, intersecting the ellipse at another point $B$. Find the maximum area of $\triangle AOB$.

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