The lac operon in Escherichia coli is a classic model of prokaryotic cellular control and gene regulation.

Which of the following statements regarding the components and regulation of the lac operon are correct?

1. The regulatory gene ($lacI$) is structurally part of the lac operon and is transcribed from the same promoter as the structural genes.
2. The binding of the inducer (allolactose) to the repressor protein alters its tertiary structure, preventing the repressor from binding to the operator.
3. The structural gene $lacY$ codes for the enzyme $\beta$-galactosidase, which catalyses the hydrolysis of lactose into glucose and galactose.