HMMT 2005 G9

Let $A C$ be a diameter of a circle $\omega$ of radius 1 , and let $D$ be the point on $A C$ such that $C D=1 / 5$. Let $B$ be the point on $\omega$ such that $D B$ is perpendicular to $A C$, and let $E$ be the midpoint of $D B$. The line tangent to $\omega$ at $B$ intersects line $C E$ at the point $X$. Compute $A X$.