Let \(Q\) be the cube with the set of vertices \(\left\{\left(x_1, x_2, x_3\right)\right\}\) \(\in R ^3: x_1, x_2, x_3\{…

Let \(Q\) be the cube with the set of vertices \(\left\{\left(x_1, x_2, x_3\right)\right\}\) \(\in R ^3: x_1, x_2, x_3\{0,1\}\). Let \(F\) be the set of all \(F\) 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

[JEE Advanced 2023]