HMMT 2007 U25

Two real numbers $x$ and $y$ are such that $8 y^{4}+4 x^{2} y^{2}+4 x y^{2}+2 x^{3}+2 y^{2}+2 x=x^{2}+1$ . Find all possible values of $x+2 y^{2}$ .

Dataset text NuminaMath olympiads. Quoted with source.

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\frac{1}{2}

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