Suppose we want to maintain a dynamic set $S$ of $n$ distinct numbers so as to support the following two operations: $(1)$ Insert a number $x$ into S$,$ and $(2)$ Report the minimum difference between two numbers in $S.$ For example, if $S=\{1,5,7,8,12\},$ then if we perform the report operation, the answer would be $1($where the minimum difference is from $7$ and $8).$

a$.$ Describe a data structure that supports each insert in $O(logn)$ time, so that each report can be performed in $O(1)$ time. Also, briefly explain why the claimed time complexities are correct.

b$.($Challenging$)$ Suppose we now need an extra operation, which deletes a number $x$ from $S.$ For example, we may delete $7$ from the previous $S$ so that $S$ becomes $\{1,5,8,12\}.$ At this point, if we perform the report operation, we should answer $3.$ Describe a data structure that supports each insert or delete in $O_{logn}$ time, while each report can still be performed in $O(1)$ time. Also, briefly explain why the claimed time complexities are correct.