If \(x_{1},x_{2},x_{3}\) are real roots of the equation \({\frac{1}{[x]}+\frac{1}{[2x]}=\{x\}+\frac{1}{3}}\), find \(x_{1}+x_{2}+x_{3}\)

Note: \([.]\) denotes greatest integer function and \(\{.\}\) denotes fractional part of \(x\).

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