(4) (30 points) Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{1}{2}, a_{n}=$ $2 a_{n} a_{n+1}+3 a_{n+1}\left(n \in \mathbf{N}^{*}\right)$.
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) If the sequence $\left\{b_{n}\right\}$ satisfies $b_{n}=1+\frac{1}{a_{n}}\left(n \in \mathbf{N}^{*}\right)$, and for any positive integer $n(n \geqslant 2)$, the inequality $\sum_{k=1}^{n} \frac{1}{n+\log _{3} b_{k}}>\frac{m}{24}$ always holds, find the maximum value of the integer $m$.