
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 its equation $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$ (where $a>b>0$) and center at O, one of its vertices is (0, 1), and its eccentricity is $\frac { \sqrt {2}}{2}$. A line passing through its right focus intersects the ellipse at points A and B.  
(1) Find the equation of this ellipse;  
(2) If OA is perpendicular to OB, find the area of triangle OAB.

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