$-$dicarbonyls are the oldest of the specific enolate equivalents, and before the

development of the use of lithium enolates, silyl enol ethers and enamines $($around the

$1950$s$-60$s$)$ they were one of the few reliable ways to alkylate a carbonyl. The scheme below

shows part of a synthesis of an isomer of menthone carried out in $1906.$

A

$($a$)$ Draw a mechanism for the conversion of $A$ to $B.$

$($b$)$ Draw a mechanism for the conversion of $C$ to $D.$

$($c$)$ Suggest reagents and a mechanism for the conversion of D to E$.$