Suppose we run the KMeans algorithm with K $=2$ on the following set of $2$D data points: $\{(2,3),(3,3),(9,5),(10,5),(11,5),(2,4),(10,6),(3,4),(9,6),(11,6)\}$ starting with initial centroids at $(2,3)$ and $(9,5).$

$1.$ Perform one iteration of the KMeans algorithm $($assignment and update steps$).$ Provide the new centroids after this iteration and assign each point to the nearest centroid.

$2.$ The KMeans algorithm aims to minimize the Within$-$Cluster Sum of Squares $($WCSS$).$ Explain the reasoning behind this objective function. What are the potential limi$-$ tations or drawbacks of this approach?

$3.$ Investigate the impact of initial centroid selection in the KMeans algorithm.

$($a$)$ Analyze at least two common strategies, discussing their pros and cons.

$($b$)$ Propose your own method for selecting initial centroids and validate its potential benefits over the existing methods that you studied in $($a$)$ using a challenging dataset of your choice.