Describe the mechanism of ring contraction of bicyclo[3.2.1]octan-2-one to bicyclo[3.1.1]heptan-2-one.

This reaction involves the ring contraction of a bicyclic ketone, specifically converting bicyclo[3.2.1]octan-2-one (an 8-carbon ketone) to bicyclo[3.1.1]heptan-2-one (a 7-carbon ketone). This transformation involves the loss of one carbon atom and a change in the bicyclic ring system.

This type of ring contraction with carbon loss from a cyclic ketone is typically achieved through a sequence involving $\alpha$-halogenation, followed by a Favorskii rearrangement, and then decarboxylation.

Step 1: $\alpha$-Halogenation
The first step is to introduce a halogen (e.g., bromine) at an $\alpha$-carbon of the ketone. For bicyclo[3.2