Let $A, B,$ and $C$ be three events in a sample space. Suppose $A$ and $B$ are mutually exclusive, $B$ and $C$ are independent, and $P(A \cap C) = 0.1$. If $P(A) = 0.3$, $P(B) = 0.4$, and $P(C) = 0.5$, find the probability of the union $P(A \cup B \cup C)$.