If $y = y(x)$ is the solution curve of the differential equation $\frac{dy}{dx} + y\tan x = x\sec x$, $0 \leq x \leq \frac{\pi}{3}$, $y(0) = 1$, then $y\left(\frac{\pi}{6}\right)$ is equal to\\
(1) $\frac{\pi}{12} - \frac{\sqrt{3}}{2}\log_e\frac{2}{e\sqrt{3}}$\\
(2) $\frac{\pi}{12} + \frac{\sqrt{3}}{2}\log_e\frac{2\sqrt{3}}{e}$\\
(3) $\frac{\pi}{12} - \frac{\sqrt{3}}{2}\log_e\frac{2\sqrt{3}}{e}$\\
(4) $\frac{\pi}{12} + \frac{\sqrt{3}}{2}\log_e\frac{2}{e\sqrt{3}}$