Given three non-empty finite sets $A, B, C$ such that $n(A) = 10, n(B) = 15, n(C) = 20$. Also, $n(A \cap B) \ge 5$, $n(B \cap C) \ge 8$, and $n(A \cap C) \ge 6$. If $n(A \cup B \cup C)$ is minimized, what is the value of $n(A \cap B \cap C)$?