
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 $C$ that shares the same foci with the hyperbola $x^{2}-y^{2}=1$ and has an eccentricity of $\frac{\sqrt{2}}{2}$.

(1) Find the standard equation of ellipse $C$.
(2) A line passing through point $P(0,1)$ intersects the ellipse at points $A$ and $B$. Point $O$ is 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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