If the shortest distance between the lines $\frac{{x - 1}}{2} = \frac{{y - 2}}{3} = \frac{{z - 3}}{4}$ and $\frac{x}{1} = \frac{y}{\alpha } = \frac{{z - 5}}{1}$ is $\frac{5}{{\sqrt 6 }}$, then the sum of all possible values of $\alpha$ is
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