Consider three sets $A, B, C$. If $n(A)=10, n(B)=12, n(C)=15$ and all intersections are empty except $n(A \cap B)=2, n(B \cap C)=3, n(A \cap C)=4$, and $n(A \cap B \cap C)=1$, find $n(A \cup B \cup C)$.