Example 1 Given that $A$ is a set of 200 different positive integers, and any 3 different numbers in $A$ are the side lengths of a non-obtuse triangle, at this time, we call this triangle a triangle determined by $A$. Let $S(A)$ denote the sum of the perimeters of all triangles determined by $A$ (congruent triangles are counted only once), find the minimum value of $S(A)$.
Given $A$, $S(A)$ is uniquely determined. To make $S(A)$ as small as possible, the elements of $A$ must be as small as possible. For this purpose, we make the smallest possible estimates for the elements of $A$.

Let $A=\left\{a_{1}, a_{2}, \cdots, a_{200}\right\}, a_{1}678^{2} \geqslant a_{k}^{2}$. Therefore, the minimum value of $S(A)$ is
$$
S\left(A_{0}\right)=\sum_{\substack{a, b, c \in A_{0} \\ a, b, c \text { ․ not equal }}}(a+b+c).
$$

For any $a \in A_{0}$, at this point $b, c$ can be any two different numbers in $A_{0} \backslash\{a\}$, with $C_{199}^{2}$ ways to choose, so $a$ is counted $C_{199}^{2}$ times in $S\left(A_{0}\right)$. By the arbitrariness of $a$, each number in $A_{0}$ is counted $C_{199}^{2}$ times in $S\left(A_{0}\right)$, so
$$
S\left(A_{0}\right)=C_{199}^{2} \sum_{a \in A_{0}} a=\frac{1}{2} \times 199 \times 198 \times \sum_{i=479}^{678} i=2279405700.
$$