You are studying the properties of a new kind of regulatory mutation of the lactose operon. This mutation, called $S$, leads to the complete repression of the $\operatorname{lac} Z$, lacY, and lacA genes, regardless of whether inducer (lactose) is present. The results of studies of this mutation in partial diploids demonstrate that this mutation is completely dominant over wild type. When you treat bacteria of the $S$ mutant strain with a mutagen and select for mutant bacteria that can express the enzymes encoded by $\operatorname{lacZ}, \operatorname{lac} Y$, and $\operatorname{lacA}$ genes in the presence of lactose, some of the mutations map to the lac operator region and others to the lac repressor gene. On the basis of your knowledge of the lactose operon, provide a molecular genetic explanation for all these properties of the $S$ mutation. Include an explanation of the constitutive nature of the "reverse mutations."