Solution (source code)

= Solution

For an \b[amine-first pathway], use <triethylamine> as a <nucleophile> in a reversible <conjugate addition>. Its <lone pair> attacks C3 of the <enone>; <electron> shifts through C2 and the <carbonyl> give a zwitterionic <enolate> with a C3–$\mathrm{NEt_3^+}$ bond.

The C2 nucleophilic site of that <enolate> attacks $\mathrm{Br_2}$, installing C2–Br and releasing <bromide>. Removal of C2–H by base then reforms C2=C3 while the C3–N bond breaks back to neutral <triethylamine>. This addition–bromination–elimination sequence gives the same \b[C2-brominated <enone>] and regenerates the nucleophilic amine, with another base equivalent accepting the eliminated <proton>.

These are alternative formal sequences requested by the question, not a kinetic determination that one must dominate under unspecified conditions. In particular, the amine-first sequence does not require <triethylamine> to generate a high concentration of a vinylic <carbanion> by direct <deprotonation>.