
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 the ellipse $\frac{y^{2}}{a^{2}}{+}\frac{x^{2}}{b^{2}}{=}1(a{ > }b{ > }0)$ with eccentricity $e{=}\frac{\sqrt{2}}{2}$ and minor axis $2b{=}2\sqrt{10}$. A parabola with its vertex at the origin is symmetric about the coordinate axes and has a focus at $(0{,}1)$.

1. Find the equations of the ellipse and the parabola.
2. Let O be the origin, A be a point in the first quadrant on the parabola, and B be a point on the ellipse such that OA is perpendicular to OB. When the midpoint of segment AB is on the y-axis, find the equation of line AB.

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