$3.(15$ points$)$ An Asynchronous Riddle

A hangman summons his $100$ prisoners, announcing that they may meet to plan a strategy, but will then be put in isolated cells, with no communication. He explains that he has set up a switch room that contains a single switch. Also, the switch is not connected to anything, but a prisoner entering the room may see whether the switch is on or off $($because the switch is up or down$).$ Every once in a while the hangman will let one arbitrary prisoner into the switch room. The prisoner may throw the switch $($on to off, or vice versa$),$ or leave the switch unchanged. Nobody but the prisoners will ever enter the switch room. The hangman promises to let any prisoner enter the room from time to time, arbitrarily often. That is$,$ eventually, each prisoner has been in the room at least once, twice, a thousand times or any number you want. At any time, any prisoner may declare: $"$We have all visited the switch room at least once." If the claim is correct, all prisoners will be released, otherwise, the hangman will execute his job $($on all the prisoners$).$

$($a$)(5$ points$)$ Which strategy would you choose if the hangman tells them, that the switch is off at the beginning?

$($b$)(5$ points$)$ Given the switch is off at the beginning as a common knowledge, if there exist $\backslash ($ f $\backslash )$ prisoners who cannot move due to their sick, how should the prisoners negotiate with the hangman?

$($c$)(5$ points$)$ Given the switch is off at the beginning as a common knowledge, if there exist $\backslash ($ f $\backslash )$ prisoners who are byzantine, what condition could the prisoners negotiate with the hangman?