Let $O$ be the center of the circumcircle of triangle $ABC$. We know that $AB = 1$ and $AO = AC = 2$. Points $D$ and $E$ lie on extensions of sides $AB$ and $AC$ beyond points $B$ and $C$ respectively, such that $OD = OE$ and $BD = \sqrt{2} \cdot EC$. Find $OD^2$.

Let's think step by step and output the final answer within \boxed{}.
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