KöMaL 2010 B4321

Prove that the following inequality is true in every triangle: $\frac{b}{\sin \left(\gamma + \dfrac{\alpha}{3}\right)} + \frac{c}{\sin \left(\beta + \dfrac{\alpha}{3}\right)} > \frac{2}{3} \cdot \frac{a}{\sin \left(\dfrac{\alpha}{3}\right)}$.