Let the line of the shortest distance between the lines \( L_1 : \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) \) and \( L_2 : \vec{r} = (4\hat{i} + 5\hat{j} + 6\hat{k}) + \mu(\hat{i} + \hat{j} - \hat{k}) \) intersect \( L_1 \) and \( L_2 \) at \( P \) and \( Q \) respectively. If \( (\alpha, \beta, \gamma) \) is the mid point of the line segment \( PQ \), then \( 2(\alpha + \beta + \gamma) \) is equal to ______.
