Let $A, B,$ and $C$ be three events in a sample space $S$. The symmetric difference of two events $X$ and $Y$, denoted by $X \Delta Y$, is the event that exactly one of the two occurs. Suppose $P(A) = 0.3$, $P(B) = 0.4$, and $P(C) = 0.2$. If $A$ and $B$ are independent, and $C$ is disjoint from $A \cup B$, determine the probability of the event $(A \Delta B) \cup C$.