已知数列 $\{a_{n}\}$ 和 $\{b_{n}\}$ 满足：$a_{1} = 1$，$a_{2} = 2$，$a_{n} > 0$，$b_{n} = \sqrt{a_{n}a_{n + 1}}$，$\ n \in \mathbf{N}^{*}$，且 $\{b_{n}\}$ 是以 $q$ 为公比的等比数列.

(1) 证明：$a_{n+2}=a_{n}q^{2}$. (2) 若 $c_{n} = a_{2n - 1} + 2a_{2n}$，证明数列 $\{c_{n}\}$ 是等比数列. (3) 求和：$\frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} + \frac{1}{a_4} + \cdots + \frac{1}{a_{2n-1}} + \frac{1}{a_{2n}}$.