If the shortest distance between the lines L_1: r =(2+λ) i +(1-3 λ) j +(3+4 λ) k, λ ∈ R; L_2: r =2(1+μ) i +3(1+μ) j +(5+μ) k, μ ∈ R is m/(√ n), where…

If the shortest distance between the lines $$\begin{array}{ll} L_1: \vec{r}=(2+\lambda) \hat{i}+(1-3 \lambda) \hat{j}+(3+4 \lambda) \hat{k}, & \lambda \in \mathbb{R} \\ L_2: \vec{r}=2(1+\mu) \hat{i}+3(1+\mu) \hat{j}+(5+\mu) \hat{k}, & \mu \in \mathbb{R} \end{array} $$ is $\displaystyle \frac{m}{\sqrt{n}}$, where $\displaystyle \operatorname{gcd}(m, n)=1$, then the value of $\displaystyle m+n$ equals