
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 $F\_1$ and $F\_2$ are the left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and $A$ is a point on the ellipse in the first quadrant such that $\overrightarrow{AF\_2} \cdot \overrightarrow{F\_1F\_2} = 0$. If the eccentricity of the ellipse is equal to $\frac{\sqrt{2}}{2}$:

1. Find the equation of the line $AO$ ($O$ is the coordinate origin);
2. The line $AO$ intersects the ellipse at point $B$. If the area of the triangle $ABF\_2$ is equal to $4\sqrt{2}$, find the equation of the ellipse.

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