
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 and a hyperbola $x^2 - y^2 = 1$ have the same foci, and the eccentricity of the ellipse is $\frac{\sqrt{2}}{2}$.  
(I) Find the standard equation of the ellipse;  
(II) A line passing through point P(0, 1) intersects the ellipse at points A and B, with O being the origin. If $\overrightarrow{AP} = 2\overrightarrow{PB}$, find the area of $\triangle AOB$.

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