记集合 $S=\left\{\left\{a_n\right\} \mid\right.$ 无穷数列 $\left\{a_n\right\}$ 中存在有限项不为零, $\left.n \in \mathbf{N}^*\right\}$, 对任意 $\left\{a_n\right\} \in S$, 设变换 $f\left(\left\{a_n\right\}\right)=a_1+a_2 x+\cdots+a_n x^{n-1}+\cdots, x \in \mathbf{R}$. 定义运算 $\otimes:$ 若 $\left\{a_n\right\},\left\{b_n\right\} \in S$, 则 $\left\{a_n\right\} \otimes\left\{b_n\right\}$ $\in S, f\left(\left\{a_n\right\} \otimes\left\{b_n\right\}\right)=f\left(\left\{a_n\right\}\right) \cdot f\left(\left\{b_n\right\}\right)$.
(1) 若 $\left\{a_n\right\} \otimes\left\{b_n\right\}=\left\{m_n\right\}$, 用 $a_1, a_2, a_3, a_4, b_1, b_2, b_3, b_4$ 表示 $m_4$;
(2) 证明: $\left(\left\{a_n\right\} \otimes\left\{b_n\right\}\right) \otimes\left\{c_n\right\}=\left\{a_n\right\} \otimes\left(\left\{b_n\right\} \otimes\left\{c_n\right\}\right)$;
(3) 若 $a_n=\left\{\begin{array}{ll}\frac{(n+1)^2+1}{n(n+1)}, & 1 \leqslant n \leqslant 100 \\ 0, & n>100\end{array}, b_n=\left\{\begin{array}{ll}\left(\frac{1}{2}\right)^{203-n}, & 1 \leqslant n \leqslant 500 \\ 0, & n>500\end{array},\left\{d_n\right\}=\left\{a_n\right\} \otimes\left\{b_n\right\}\right.\right.$,
证明: $d_{200} < \frac{1}{2}$.