$[36$ pts$]$ Consider the following eight two$-$dimensional data points:

x$1(15,10),$ x$2(3,10),$ x$3(15,12),$ x$4(3,14),$ x$5(18,13),$ x$6(1,7),$ x$7(10,1),$ x$8(10,30)$

You are required to apply the k$-$means algorithm to cluster these points using the Euclidean distance. You need to show the information about each final cluster $($including the mean of the cluster and all data points in this cluster$).$

$($a$)[8$ pts$]$ If k $=2$ and the initial means are $(10,1)$ and $(10,30),$ what is the output of the algorithm?

$($b$)[8$ pts$]$ If k $=3$ and the initial means are $(10,1),(10,30),$ and $(3,10),$ what is the output of the algorithm?

$($c$)[8$ pts$]$ If k $=4$ and the initial means are $(10,1),(10,30),(3,10),$ and $(15,10),$ what is the output of the algorithm?

$($d$)[12$ pts$]$ What are the advantages and disadvantages of algorithm k$-$means? For each disadvantage, please also give a suggestion to enhance the algorithm.