A circle with center $C$ passes through point $B$. Let $A$ be a point outside of the circle and $Q$ be the intersection of the circle and $\overline{AB}$. If $\angle ACB$ is a right angle and $\overline{AB}$, $\overline{AC}$, $\overline{BC}$, and $\overline{BQ}$ have integer lengths, find the minimum possible perimeter of $\triangle CBQ$.