Consider a Chord P$2$P system that uses $4-$bit identifiers $\backslash (($m$=4)\backslash ).$ This means that node IDs belong to the range of $0$ to $15.$ Suppose the current nodes in the system have the following IDs: $2,6,10,13,14.$

a$)$ What is the successor node for key $12?$

b$)$ Identify which nodes store the following keys: $\backslash (0,1,4,6,9,13,15\backslash ).$

c$)$ Compute the finger tables for nodes $\backslash (6,10,13,14\backslash ).$

d$)$ Suppose that node $\backslash (\backslash$mathbf$\{2\}\backslash )$ is looking for the node that stores the file with key $12.$ Show how this lookup request will be forwarded from node $2$ to the correct node, including the intermediate steps and nodes involved in the process.

e$)$ Construct the multicast tree with mid $=9$ as the multicast identifier randomly generated by the initiator and nodes $13,14,2,$ and $6$ joining in sequence. Show the tree structure and describe how messages would be propagated through the system.