$2.(24$ points$)$ Using the following $\backslash (\backslash$mathrm$\{$B$\}+\backslash )$ Tree index, answer the questions that follow: $($show all work$)$

$-$ Every node can contain $\backslash ($ m $\backslash )$ entries, where $\backslash (2\backslash$leq m $\backslash$leq $4\backslash )$

$-$ The root node is an exception: it allows $\backslash (1\backslash$leq $\backslash$mathrm$\{$m$\}\backslash$leq $4\backslash )$

I. Show the resulting B$+$Tree after deleting data entry $42*$

II$.$ Using the tree derived from part $($I$)$ above, show the resulting B$+$Tree after inserting data entry $7*$

III. Using the tree derived from part $($II$)$ above, show the resulting B$+$Tree after inserting data entry $\backslash (31*\backslash )$

Draw separate B$+$ tree for I, II$,$ and III and describe the steps $($Show all work$)$