设数列 $\{a_{n}\}$ 满足 $a_{1} = 2$，$a_{2} + a_{4} = 8$，且对任意 $n \in \mathbf{N}^{*}$，函数 $f(x) = (a_{n} - a_{n+1} + a_{n+2})x + a_{n+1}\cos x - a_{n+2}\sin x$ 满足 $f'\!\left(\frac{\pi}{2}\right) = 0$.

(1) 求数列 $\{a_{n}\}$ 的通项公式； (2) 若 $b_{n} = 2\left(a_{n} + \frac{1}{2a_{n}}\right)$，求数列 $\{b_{n}\}$ 的前 $n$ 项和 $S_{n}$.