Design a data structure for the following operations:

$($a$)$ Insert$($S$,$ x$)$: Insert x into S in O$($log n$)$ time.

$($b$)$ Search$($x$)$: Search for x in S in O$($log n$)$ time.

$($c$)$ Delete$($S$,$ x$)$: Delete x from S$.$ In O$($log n$)$ time.

$($d$)$ M ax $$ Gap$($S$)$: find the maximum difference, maxx,y in S $\{|$x $$ y$|\},$ between two

elements of S in time O$(1)$ time.

$($e$)$ M in $$ Gap$($S$)$: find the two numbers x$,$ y in S such that $|$x $$ y$|$ is the smallest,

i$.$e$.,$ the difference between the closest two numbers: minx,y in S$,$x$=$y$\{|$x$$y$|\},$ in O$(1)$

time.