
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 a point P(x, y) on the ellipse C (not including points on the x-axis) that satisfies the following condition: the product of the slopes of the lines connecting P to points A($-\sqrt{2}$, 0) and B($\sqrt{2}$, 0) is equal to $-1/2$. Points A and B also lie on the curve C.

1. Find the equation of the ellipse C.
2. Draw a line passing through the right focus of ellipse C with a slope of 1, intersecting the ellipse at points M and N. Find the length of the line segment |MN|.
3. Find the minimum distance from a point on the ellipse to the line $\sqrt{2}x + y + 2\sqrt{5} = 0$.

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