Laplace Transforms · Superchallenge problem

Find the function $y(t)$, positive for small $t > 0$ and having a Laplace transform, that satisfies $\displaystyle\int_0^t y(\tau)\,y(t - \tau)\,d\tau = \dfrac{e^{2t}}{54}\left(\sin 3t - 3t\cos 3t\right)$ for $t \ge 0$. Give $y\!\left(\dfrac{\pi}{6}\right)$.