
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 the center of symmetry at the origin, one of its foci coincides with the center of the circle $x^{2}+y^{2}-2 \sqrt {2}x=0$, and the ellipse passes through the point $( \sqrt {2},1)$.  
$(1)$ Find the standard equation of the ellipse;  
$(2)$ A line passing through point $P(0,1)$ intersects the ellipse at points $A$ and $B$, with $O$ as the coordinate 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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