Three Dimensional Geometry: JEE Main 2025 Mathematics question

Let the values of $p$, for which the shortest distance between the lines $\displaystyle\frac{x+1}{3}=\frac{y}{4}=\frac{z}{5}$ and $\mathbf{r}=(p\hat{i}+2\hat{j}+\hat{k})+\lambda(2\hat{i}+3\hat{j}+4\hat{k})$ is $\tfrac{1}{\sqrt6}$, be $a,b$ with $a<b$. Then the length of the latus rectum of the ellipse $\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is: