Recall the ideas explored in Homework Problems$-$I and its solution.

In Problem $8($Homework Problems$-$I$),$ we recognized the possible inefficient use of a $w-$unit wide doorway, where $w>2.$ A key reason for this was the fact that the robots, that are placed in a straight line, could not tell how many robots there were or determine the part of the door to move to in a collision$-$free manner.

If robots were transparent, that is every robot can see all robots, then they can systematically take turns to move $w$ robots at a time through distinct parts of the door.

We now recall some details of the problem doorway $(n,w)$ defined in Homework Problems$-$I. The points robots operate on a $ZZ$ plane $($grid$)$ starting and stopping only at grid points. The door of width $w$ is parallel to one of the grid axes $($say the horizontal axis, as in Figure $2$ of Homework Problems$-$I$).$ The robots are placed on distinct grid points in a straight line parallel to the door. The robots have lights and run in Look$-$Compute$-$Move $($LCM$)$ steps.

As outlined in Problems $9,10$ of Homework Problems$-$I, one of the end robots can be identified quickly $($on an average$)$ as the leader with a special light color.

$11.$ You are given a swarm of $n$ transparent robots placed in a line with one of the end robots as the leader. Initially, the leader has a blue light and other robots have green lights.

Write a $($set of$)$ condition$-$action pairs that solve $drway(n,w)$ optimally in $|$~$\frac{n}{w}$~$|$ steps. Explain why your algorithm works $($including without collision$),$ and in the above number of steps.