HMMT 2006 K9

Compute the sum of all real numbers $x$ such that

$2 x^{6}-3 x^{5}+3 x^{4}+x^{3}-3 x^{2}+3 x-1=0$

Dataset text NuminaMath olympiads. Quoted with source.

Show the answer

-\frac{1}{2}

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