$[30$ points$]$ Implement K$-$means manually.

Figure $1$: Scatter plot of datasets and the initialized centers of $3$ clusters

Given the input matrix $x$ whose rows represent different data points, please perform a k$-$means

clustering on this dataset using the Euclidean distance as the distance function. Here $k$ is chosen

as $3.$ The Euclidean distance $d$ between input vector vec$(p)in{R}^n$ and vec$(q)in{R}^n$ is defined as

$d=\sqrt[\phantom{2}]{{\displaystyle \underset{i=1}{\overset{n}{}}}(p_i-q_i{)}^2}$

All data in $x$ were plotted in Figure $1.$ The center of $3$ clusters were initialized as vec$(c_1)=(6\mathrm{.}2,3\mathrm{.}2)$

$($red$),$ vec$(c_2)=(6\mathrm{.}6,3\mathrm{.}7)($green$),$ vec$(c_3)=(6\mathrm{.}5,3\mathrm{.}0)($blue$).$