Unclassified (maths): JEE Advanced 2023 Mathematics question

Let $Q$ be the cube with the set of vertices $\left\{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}: x_{1}, x_{2}, x_{3} \in\{0,1\}\right\}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ be the set of all four lines containing the main diagonals of the cube $Q$; for instance, the line passing through the vertices $(0,0,0)$ and $(1,1,1)$ is in $S$. For lines $\ell_{1}$ and $\ell_{2}$, let $d\left(\ell_{1}, \ell_{2}\right)$ denote the shortest distance between them. Then the maximum value of $d\left(\ell_{1}, \ell_{2}\right)$, as $\ell_{1}$ varies over $F$ and $\ell_{2}$ varies over $S$, is