设等差数列 $\left\{a_{n}\right\}$ 的前 $n$ 项和为 $S_{n}, a_{3}=4, a_{4}=S_{3}$, 数列 $\left\{b_{n}\right\}$ 满足:
对每个 $n \in \mathbf{N}^{*}, S_{n}+b_{n}, S_{n+1}+b_{n}, S_{n+2}+b_{n}$ 成等比数列.
(1) 求数列 $\left\{a_{n}\right\},\left\{b_{n}\right\}$ 的通项公式;
(2) 记 $C_{n}=\sqrt{\frac{a_{n}}{2 b_{n}}}, n \in \mathbf{N}^{*}$, 证明: $C_{1}+C_{2}+\mathrm{L}+C_{n} < 2 \sqrt{n}, n \in \mathbf{N}^{*}$.
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