HMMT 2015 G7

Let $A B C D E$ be a square pyramid of height $\frac{1}{2}$ with square base $A B C D$ of side length $A B=12$ (so $E$ is the vertex of the pyramid, and the foot of the altitude from $E$ to $A B C D$ is the center of square $A B C D$ ). The faces $A D E$ and $C D E$ meet at an acute angle of measure $\alpha$ (so that $0^{\circ}<\alpha<90^{\circ}$ ). Find $\tan \alpha$.