A division of a group of people into various groups is called $k$-regular if the number of groups is less or equal to $k$ and two people that know each other are in different groups. Let $A$, $B$, and $C$ groups of people such that there are is no person in $A$ and no person in $B$ that know each other. Suppose that the group $A \cup C$ has an $a$-regular division and the group $B \cup C$ has a $b$-regular division. For each $a$ and $b$, determine the least possible value of $k$ for which it is guaranteed that the group $A \cup B \cup C$ has a $k$-regular division.