
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 centered at the origin with foci on the x-axis, and the quadrilateral formed by the endpoints of the ellipse's minor axis and the foci is a square. The maximum distance from a point on the ellipse to a focus is $\sqrt{2} + 1$.
1. Find the standard equation of the ellipse.
2. A line passing through the left focus F of the ellipse intersects the ellipse at points A and B, and the midpoint of segment AB lies on the line $x+y=0$. Find the equation of line AB.

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