HMMT 2013 G9

Pentagon $A B C D E$ is given with the following conditions: (a) $\angle C B D+\angle D A E=\angle B A D=45^{\circ}, \angle B C D+\angle D E A=300^{\circ}$ (b) $\frac{B A}{D A}=\frac{2 \sqrt{2}}{3}, C D=\frac{7 \sqrt{5}}{3}$, and $D E=\frac{15 \sqrt{2}}{4}$ (c) $A D^{2} \cdot B C=A B \cdot A E \cdot B D$

Compute $B D$.