Let the function $f(x) = \cos(\omega x + \varphi)$ ($\omega > 0, 0 < \varphi < \pi$) have minimum positive period $T$. If $f(T) = \frac{\sqrt{3}}{2}$ and $x = \frac{\pi}{6}$ is a zero of $f(x)$, then the minimum value of $\omega$ is $\_\_\_\_$.
Let the function $f(x) = \cos(\omega x + \varphi)$ ($\omega > 0, 0 < \varphi < \pi$) have minimum positive period $T$. If $f(T) = \frac{\sqrt{3}}{2}$ and $x = \frac{\pi}{6}$ is a zero of $f(x)$, then the minimum value of $\omega$ is $\_\_\_\_$.