Example 16 (Mathematical Olympiad Problems from "Intermediate Mathematics" Issue 4, 2004) Try to find the maximum number of elements in a set $S$ that satisfies the following conditions:
(1) Each element in $S$ is a positive integer not exceeding 100;
(2) For any two different elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the greatest common divisor of $a+b$ and $c$ is 1;
(3) For any two different elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the greatest common divisor of $a+b$ and $c$ is greater than 1.