HMMT 2015 G4

Let $A B C D$ be a cyclic quadrilateral with $A B=3, B C=2, C D=2, D A=4$. Let lines perpendicular to $\overline{B C}$ from $B$ and $C$ meet $\overline{A D}$ at $B^{\prime}$ and $C^{\prime}$, respectively. Let lines perpendicular to $\overline{A D}$ from $A$ and $D$ meet $\overline{B C}$ at $A^{\prime}$ and $D^{\prime}$, respectively. Compute the ratio $\frac{\left[B C C^{\prime} B^{\prime}\right]}{\left[D A A^{\prime} D^{\prime}\right]}$, where $[\varpi]$ denotes the area of figure $\varpi$.