(20 (12 points) Given the function $f(x)=\frac{2 x+3}{3 x}$, the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{n+1}=$ $f\left(\frac{1}{a_{n}}\right), n \in \mathbf{N}^{*}$.
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) Let $T_{n}=a_{1} a_{2}-a_{2} a_{3}+a_{3} a_{4}-a_{4} a_{5}+\cdots-a_{2 n} a_{2 n+1}$, find $T_{n}$.