PLease respond a through f $[25$ points$]$ Consider a system of five processes that implements the Paxos algorithm for

consensus. As shown in the figure below, P$1$ and P$2$ are proposers. P$3,$ P$4,$ P$5$ are acceptors. P$1$

sends a prepare message with proposal #$2$ to processes P$3$ and P$5$ and receives promise replies

from them. P$1$ then sends a propose message with proposed value $20$ for proposal #$2.$ P$2$

concurrently sends a prepare message with proposal #$6,$ with an initial intention to propose a

value of $25$ if it receives sufficient replies. Only a subset of responses from processes P$3$ and P$5$

are shown in the figure. Assume no other proposals are initiated. Answer the following questions.

a$.$ Which processes will accept P$1\text{'}$s proposal?

b$.$ Which processes will reply back to P$2\text{'}$s prepare message? For each such process, specify the

message $($including its parameters$)$ that it will send back to P$2.$

c$.$ Will P$2$ send out a propose message for its proposal #$6?$ If yes, what value will it propose?

d$.$ Consider the state of the system at time $11.$ Has the proposed value $20$ been implicitly

decided upon at this point? If yes, explain why. If not, construct a possible scenario that may

occur after time $11$ where the system ends up deciding on a different proposed value.

Illustrate this scenario by drawing a figure similar to the one above. The scenario would

involve a temporal sequence of events that may differ from the one shown in the figure

beyond time $11$ but must comply with what is shown until time $11.$

e$.$ Suppose that P$1\text{'}$s propose message reaches $P3$ at time $9($instead of time $15).$ If we now

consider the state of the system at time $11,$ has the proposed value of $20$ been implicitly

decided upon? Explain why.

f$.$ Suppose that P$1\text{'}$s propose message reaches P$5$ at time $16($instead of time $10)$ and reaches P$3$

at the original time $15.$ Will P$2$ send out a propose message for its proposal #$6?$ If yes, what

value will it propose?