Three. (20 points) For a set $S\left(S \subseteq \mathbf{N}_{+}\right)$, if for any $x \in S, \dot{x}$ cannot divide the sum of elements of any non-empty subset of $S \backslash\{x\}$, then $S$ is called a "good set" ($S \backslash\{x\}$ represents the set $S$ after removing the element $x$).
(1) If $\{3,4, n\}$ is a good set, find the minimum value $n_{0}$ of $n$;
(2) Prove: the set $\left\{3,4, n_{0}, m\right\}$ is not a good set, where $m \in \mathbf{N}_{+}, m \neq 3,4, n_{0}$.
Three, (1) Obviously, $n>4$.
If $n=5$, then $4 \mid (3+5)$; if $n=6$, then $3 \mid 6$; if $n=7$, then $7 \mid (3+4)$; if $n=8$, then $4 \mid 8$; if $n=9$, then $3 \mid 9$.
When $n=10$, it is verified that $\{3,4,10\}$ is a good set.
Therefore, the minimum value of $n$ is $n_{0}=10$.
(2) If $\{3,4,10, m\}$ is a good set, then $m$ is not a multiple of 3, and 3 cannot divide $10+m$, i.e., $m$ is not of the form $3k+2$. 3 also cannot divide $10+4+m$, i.e., $m$ is not of the form $3k+1$. Therefore, for any positive integer $m (m \neq 3,4,10)$, the set $\{3,4,10, m\}$ is not a good set.