# Maximize and Minimize?

Geometry Level 3

A square $$ABCD$$ of unit area is given.

A quadrilateral $$EFGH$$ is drawn such that $$E$$ lies on $$AB$$, $$F$$ lies on $$BC$$, $$G$$ lies on $$CD$$ and $$H$$ lies on $$DA$$.

If the minimum and maximum possible values for the sum of the squares of the lengths of the sides of $$EFGH$$ are $$\alpha$$ and $$\beta$$ respectively, find the value of:

$\alpha^{\beta} + \beta^{\alpha}$

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