Let $DE$ the diameter of a circunference $\Gamma$. Let $B, C$ on $\Gamma$ such that $BC$ is perpendicular to $DE$, and let $Q$ the intersection of $BC$ with $DE$. Let $P$ a point on segment $BC$ such that $BP=4PQ$. Let $A$ the second intersection of $PE$ with $\Gamma$. If $DE=2$ and $EQ=\frac{1}{2}$, find all possible values of the sides of triangle $ABC$.