If $\alpha, \beta, \gamma$ are the three roots of $x^{3}-x+1$, determine the value of

$$
\frac{1-\alpha}{1+\alpha}+\frac{1-\beta}{1+\beta}+\frac{1-\gamma}{1+\gamma}
$$
Given that $x$ is a root of $x^{3}-x+1$, we wonder about the polynomial for which $y=\frac{1-x}{1+x}$ is a root. We find $x=\frac{1-y}{1+y}$ and substitute it into the polynomial. We then find that $y$ is a root of $-y^{3}+y^{2}-7 y-1$. By Viète's formulas, we conclude that the desired sum is 1.