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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