Three Dimensional Geometry: JEE Advanced 2023 Mathematics question

Let $\ell_{1}$ and $\ell_{2}$ be the lines $\vec{r}_{1}=\lambda(\hat{i}+\hat{j}+\hat{k})$ and $\vec{r}_{2}=(\hat{j}-\hat{k})+\mu(\hat{i}+\hat{k})$, respectively. Let $X$ be the set of all the planes $H$ that contain the line $\ell_{1}$. For a plane $H$, let $d(H)$ denote the smallest possible distance between the points of $\ell_{2}$ and $H$. Let $H_{0}$ be a plane in $X$ for which $d\left(H_{0}\right)$ is the maximum value of $d(H)$ as $H$ varies over all planes in $X$. Match each entry in List-I to the correct entries in List-II. \textbf{List-I} (P) The value of $d\left(H_{0}\right)$ is (Q) The distance of the point $(0,1,2)$ from $H_{0}$ is (R) The distance of origin from $H_{0}$ is (S) The distance of origin from the point of intersection of planes $y=z, x=1$ and $H_{0}$ is \textbf{List-II} (1) $\sqrt{3}$ (2) $\frac{1}{\sqrt{3}}$ (3) 0 (4) $\sqrt{2}$ (5) $\frac{1}{\sqrt{2}}$ The correct option is: