5. For any set $S$, let $|S|$ denote the number of elements in the set, and let $n(S)$ denote the number of subsets of set $S$. If $A, B, C$ are three sets that satisfy the following conditions:
(1) $n(A)+n(B)+n(C)=n(A \cup B \cup C)$;
(2) $|A|=|B|=100$.

Find the minimum value of $|A \cap B \cap C|$.
5. If a set has $k$ elements, then it has $2^{k}$ subsets, and $2^{100}+2^{100}+2^{|c|}=2^{|A \cup B \cup C|}, 1+2^{|c|-101}=2^{|A \cup B \cup C|-101}$.
Since $1+2^{|c|-101}$ is greater than 1 and equals an integer power of 2, $|C|=101$, so, $|A \cup B \cup C|=102$.
By the principle of inclusion-exclusion, $|A \cap B \cap C|=403-|A \cup B|-|\dot{A} \cup C|-|B \cup C|$.
From $A \cup B, A \cup C, B \cup C \subseteq A \cup B \cup C$ we get $|A \cup B|,|B \cup C|, |C \cup A| \leqslant 102$.
Therefore, $|A \cap B \cap C|=403-|A \cup B|-|A \cup C|-|B \cup C| \geqslant 97$.
For example: $A=\{1,2,3, \cdots, 100\}, B=\{3,4,5, \cdots, 102\}, C=\{1,2,4,5,6, \cdots, 101,102\}$, thus $|A \cap B \cap C|=|\{4,5,6, \cdots, 100\}|=97$, which is possible.