= Solution
The first step is a \b[vinylogous <aldol condensation> at C5 of the unsaturated <lactam>]. Number <nitrogen> N1, the <lactam> <carbonyl> C2, and then the methoxy-substituted <alkene> C3=C4 and the <methylene group> C5.
<hydroxide> reversibly removes a C5 <proton>. The resulting <vinylogous enolate> is delocalized through C5–C4–C3–C2–O: it is not necessary to create an unstabilized isolated <carbanion> beside <nitrogen>. Its C5 nucleophilic contributor attacks acetone. <protonation> gives a C5-linked tertiary <alcohol>, $\mathrm{C5-C(OH)(Me)_2}$. Further removal of the remaining C5 <hydrogen> and elimination through an aldol-dehydration sequence forms \b[C5=C(Me)2], leaving the original methoxy <enol ether> in place.
In the second step, \b[aqueous acid hydrolyses that <enol ether> to a <ketone> at C4]. <protonation> at C3 gives an oxygen-stabilized cation at C4. <water> attacks C4, <proton transfers> form a hemiacetal-like intermediate, and loss of <methanol> followed by <deprotonation> produces C4=O and the C3 <methylene group>.
The <lactam> <carbonyl> and N–H group remain, as does the exocyclic isopropylidene group. This accounts for replacement of the C3=C4–OMe unit by C3H2–C4=O without inserting a new <oxygen> into the ring skeleton.
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