If $\alpha$, $\beta$, and $\gamma$ are the roots of the equation $x^3 - x - 1 = 0$, compute the expression $\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}$.

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