JEE Advanced 2023 Mathematics question on Three Dimensional Geometry

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 $d(H_0)$ 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.
List-I
(P) The value of $d(H_0)$ 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
List-II
(1) $\sqrt{3}$
(2) $\dfrac{1}{\sqrt{3}}$
(3) $0$
(4) $\sqrt{2}$
(5) $\dfrac{1}{\sqrt{2}}$