First, evaluate \(-\cos^{-1}\left(\frac{1}{2}\sqrt{3}\right)\). The value of \(\cos^{-1}\left(\frac{1}{2}\sqrt{3}\right)\) is \(\frac{\pi}{6}\), so \(-\cos^{-1}\left(\frac{1}{2}\sqrt{3}\right) = -\frac{\pi}{6}\).

Substitute this into the equation:

\(\frac{1}{6}\pi + \arctan(4x) = -\frac{\pi}{6}\).

Rearrange to find \(\arctan(4x)\):

\(\arctan(4x) = -\frac{\pi}{6} - \frac{1}{6}\pi = -\frac{\pi}{3}\).

Thus, \(4x = \tan\left(-\frac{\pi}{3}\right) = -\sqrt{3}\).

Solving for \(x\), we get:

\(x = -\frac{\sqrt{3}}{4}\).