If the shortest distance between the lines;
\( L_{1} : \ \vec{r} = (2 + \lambda)\hat{i} + (1 - 3\lambda)\hat{j} + (3 + 4\lambda)\hat{k}, \ \lambda \in \mathbb{R} \)
\( L_{2} : \ \vec{r} = 2(1 + \mu)\hat{i} + 3(1 + \mu)\hat{j} + (5 + \mu)\hat{k}, \ \mu \in \mathbb{R} \)
is \( \dfrac{m}{\sqrt{n}} \), where \( \gcd(m, n) = 1 \), then the value of \( m + n \) equals:
