
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
Let the line \( y = x + \sqrt{2} \) intersect the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) (where \( a > b > 0 \)) at points \( M \) and \( N \), and suppose \( OM \perp ON \) (where \( O \) is the origin). If \( MN = \sqrt{6} \), find the equation of the ellipse.

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