Let $A B C D$ be a square of side of unit length. Let a circle $C_1$ centered at $A$ with unit radius is drawn. Another circle $C_2$ which touches $C_1$ and the lines $A D$ and $A B$ are tangent to it, is also drawn. Let a tangent line from the point $C$ to the circle $C_2$ meet the side $A B$ at $E$. If the length of $E B$ is $\alpha+\sqrt{3} \beta$, where $\alpha, \beta$ are integers, then $\alpha+\beta$ is equal to $\_\_\_\_$ .