Graphs of Sine and Cosine · Superchallenge problem

Let $f(x)=\sin(\omega x+\varphi)$ with $\omega>0$ and $|\varphi|\le\dfrac{\pi}{2}$. The point $x=-\dfrac{\pi}{3}$ is a zero of $f$, the vertical line $x=\dfrac{\pi}{6}$ is an axis of symmetry of its graph, and $f$ is monotonic (only increasing or only decreasing) on the interval $\dfrac{\pi}{5}<x<\dfrac{3\pi}{10}$. Find the largest possible value of $\omega$.