$($c$)$ In fact, Compare can be improved to use fewer comparisons. Give a new algorithm Compare$2$ that gives the same output as Compare but uses fewer comparisons between elements. Write down the exact number of element comparisons made by Compare$2.($Hint: after comparing max $1$ and max$2,$ you already know that one of the remaining sndmax $1$ and sndmax $2$ cannot be the second maximum.$)[6$ marks$]($d$)$ Suppose now we replace Compare with Compare$2$ in the algorithm MaxMax. Write down the recurrence expressing the time complexity of this new version of MaxMax $($count only the number of comparisons between elements of S $).$ Again solve this new recurrrence using Part $($a$).[4$ marks$]$