Question 240, Set $S$ satisfies the following conditions: (1) The elements in $S$ are all positive integers not exceeding 100; (2) For any $a, b \in S, a \neq b$, there exists a $c \in S$ different from $a, b$, such that $(a+b, c)=1$; (3) For any $a, b \in S, a \neq b$, there exists a $c \in S$ different from $a, b$, such that $(a+b, c)>1$. Try to find the maximum value of $|S|$.

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Note: The notation $(a, b)$ represents the greatest common divisor (gcd) of $a$ and $b$.
Question 240, Solution: (1) First prove: $|S| \leq 50$.
If $|S| \geq 51$, then there must be two distinct elements in $S$ whose sum is 101. Let these two elements be $a$ and $b$. Then for any $c \in S$, we have $(a+b, c) = (101, c) = 1$. This contradicts condition (3).
(2) Next prove: $S = \{2\} \cup \{1, 3, 5, \ldots, 97\}$ satisfies the conditions.
(1) For any $a, b \in S$, if $a$ and $b$ do not contain 1, then $(a+b, 1) = 1$; if $a$ and $b$ contain 1 but not 97, then $(a+b, 97) = 1$; if $a$ and $b$ are 1 and 97 respectively, then $(a+b, 95) = 1$.
(2) For any $a, b \in S$, if $a$ and $b$ do not contain 2, then $(a+b, 2) = 2 > 1$.
If $a$ and $b$ contain 2, assume $a = 2$. If $b \neq 97$, then $(a+b, b+2) = b+2 > 1$; if $b = 97$, then $(a+b, 3) = 3 > 1$.
In summary, the maximum value of $|S|$ is 50.