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

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