
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 line \( x + y = 1 \) intersects the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) at points \( A \) and \( B \). \( C \) is the midpoint of the line segment \( AB \). If \( |AB| = 2 \sqrt{2} \) and the slope of the line \( OC \) is \( k_C = \frac{1}{\sqrt{2}} \), find the equation of the ellipse.

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