
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
The ellipse $ax^2+by^2=1$ (where $a>0, b>0, \text{and } a \neq b)$ intersects the line $x+y-1=0$ at points A and B. Let C be the midpoint of AB. Given $|AB|=2\sqrt{2}$, and the slope of the line OC is $\frac{\sqrt{2}}{2}$, find the equation of the ellipse.

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