Consider the Closest Pair problem using divide$-$and$-$conquer covered in class. Now, assume that this problem is operating in a dynamic environment, where new points can be inserted, and existing points can be deleted at any time.

Design an efficient algorithm that supports both insertion and deletion operations while continuously maintaining the closest pair of points after each update. The brute$-$force method, which involves re$-$running the entire closest pair algorithm whenever a point is added or removed, is highly inefficient. Your objective is to create an algorithm that minimizes the need for recomputation and handles these dynamic updates efficiently.

$($a$)$ What is the time complexity of your algorithm for both the insert and delete operations in all possible cases?

$($b$)$ Calculate the exact number of operations required for inserting or deleting a point in the dataset for each possible case.