Consider the following computation of a decentralized basic algorithm on an undirected ring of size three, with processes p$,$ q and r$,$ where p and q are the initiators. First, p sends a message to q and r and becomes passive, while q sends a message to r$.$ When q receives p$$s message, it also becomes passive. After reception of the messages from first p and then q$,$ r sends a message to both p and q and becomes passive. After reception of the message from r$,$ p and q send a message to each other, and after reception of these messages become passive. Add the following termination detection algorithms on top of the basic algorithm, and in each case extend the computation with control messages, to explain how termination is detected.

$($a$)$ The Shavit$-$Francez algorithm.

$($b$)$ Rana$$s algorithm.

$($c$)$ Safra$$s algorithm, with p as initiator of this control algorithm.