12. Given the sequence $\left\{a_{n}\right\}$, the sum of the first $n$ terms is $S_{n}=-n^{2}+9 n+2\left(n \in \mathbf{N}^{*}\right)$.
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) Let $b_{n}=\frac{1}{n\left(12-a_{n}\right)}\left(n \in \mathbf{N}^{*}\right), T_{n}=b_{1}+b_{2}+\cdots+b_{n}$, does there exist a smallest positive integer $m$, such that the inequality $T_{n}<\frac{m}{32}$ holds for all positive integers $n$? If it exists, find the value of $m$; if not, explain the reason.