已知数列 $\left\{a_n\right\}$ 满足 $a_n \neq 0,\left(1+3 a_1\right)\left(1+3 a_2\right)\left(1+3 a_3\right) \cdots\left(1+3 a_n\right)=a_n\left(n \in N^*\right)$.
(1)证明: 数列 $\left\{\frac{1}{a_n}\right\}$ 是等差数列;
(2)求数列 $\left\{a_n a_{n+1}\right\}$ 的前 $n$ 项和 $T_n$.
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