
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 $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, its center is $O$, 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 $OA \perp OB$, find the area of $\triangle OAB$.

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