$(15\%)$ In Chapter $4,$ we learned balanced binary search trees.

Consider the insertion of items with the following $9$ keys $($in the given order$)$ into an initially empty AVL tree: $13,18,29,84,52,67,20,9,37,90,$ Show each step of insertions and necessary rotations. For each rotation, according to Algorithm $4\mathrm{.}2\mathrm{.}2$ and Figure $4\mathrm{.}2\mathrm{.}3,$ explain which case applied and what are $\frac{a}{b}\frac{?}{c}$ and $\frac{x}{y}\frac{?}{z}.$

Convert this AVL tree to be a red$-$black tree.