Existence, Uniqueness & Bifurcations · Superchallenge problem

Treat $y$ as an angle: $y' = \dfrac53 - \sin y$ with $y(0) = 0$. Find $\displaystyle\lim_{T\to\infty}\dfrac1T\int_0^T \sin y(t)\,dt$. Express your answer as a common fraction.