For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$, and $C$ are sets for which 
\[n(A) + n(B) + n(C) = n(A \cup B \cup C)\] 
and 
\[|A| = |B| = 100,\] 
then find the minimum possible value of $|A \cap B \cap C|$.

Let's think step by step and output the final answer within \boxed{}.
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