Suppose we have an array A $=[$k$,$ d$,$ c$,$ i$,$ t$,$ e$,$ s$,$ n$,$ o$,$ p$,$ j$,$ u$,$ x$,$ a$,$ q$,$ y$,$ l$,$ b$,$ g$,$ m$,$ h$,$ r$,$ v$,$ f$,$ w$]$ which includes $25$ integers and we are trying to find the median of A$.$ You can assume that variables that are lexicographically greater have greater values.

$($a$)$ Utilize the median of medians algorithm to find the "true" median of A$.$ Please use subarrays of size $5.$ If the length of the array is not a multiple of $5,$ the last sub$-$array can be incomplete. Note that you do not have to show work when sorting sub$-$arrays or ordering the medians of sub$-$arrays.

$($b$)$ In the first iteration, what values are guaranteed to be less than the median of medians? What values are guaranteed to be greater than the median of medians?

$($c$)$ What would be the time complexity of this algorithm if our sub$-$arrays were of size $10?$