Given the function $f(x) = \cos(2x + \varphi)$ $(0 \leq \varphi < \pi)$, $f(0) = \frac{1}{2}$. (1) Find $\varphi$; (2) Let $g(x) = f(x) + f\left(x - \frac{\pi}{6}\right)$. Find the range and monotonic intervals of $g(x)$.
Given the function $f(x) = \cos(2x + \varphi)$ $(0 \leq \varphi < \pi)$, $f(0) = \frac{1}{2}$.\\
(1) Find $\varphi$;\\
(2) Let $g(x) = f(x) + f\left(x - \frac{\pi}{6}\right)$. Find the range and monotonic intervals of $g(x)$.