17 (10 points) Given the quadratic function $f(x)=x^{2}-t x+t$ where $t \in \mathbf{R}$, and it satisfies: (1) The solution set of the inequality $f(x) \leqslant 0$ has exactly one element; (2) If $0<x_{2}<x_{1}$, then the inequality $f\left(x_{2}\right)<f\left(x_{1}\right)$ always holds. Let the sequence $\left\{a_{n}\right\}$ have the sum of its first $n$ terms $S_{n}=f(n)$.
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) Find $\lim _{n \rightarrow \infty}\left(\frac{1}{a_{1} a_{3}}+\frac{1}{a_{2} a_{4}}+\frac{1}{a_{3} a_{5}}+\cdots+\frac{1}{a_{n} a_{n+2}}\right)$.