The largest value of \( a \), for which the perpendicular distance of the plane containing the lines \( \vec{r} = (\hat{i}+\hat{j}) + \lambda(\hat{i}+a\hat{j}-\hat{k}) \) and \( \vec{r} = (\hat{i}+\hat{j}) + \mu(-\hat{i}+\hat{j}-a\hat{k}) \) from the point \( (2,1,4) \) is \( \sqrt{3} \), is ______.
