Let $A$, $B$, and $C$ be three events in a sample space. Suppose $A$ and $B$ are independent, $B$ and $C$ are mutually exclusive, and $A$ and $C$ are independent. If $P(A) = 0.4$, $P(B) = 0.3$, and $P(C) = 0.2$, what is the probability $P(A \cup B \cup C)$?