MathNet 2014 Question 51

Let $ABCD A_1 B_1 C_1 D_1$ be a rectangular cuboid with edge-lengths $|AB| = |AD| = a$ and $|AA_1| = 2a$. Let $A', B', C'$ and $D'$ be the midpoints of $AA_1, BB_1, CC_1$ and $DD_1$ respectively.
Find the volume of the intersection of the cube $ABCD A'B'C'D'$ and the pyramid $A_1A'B'B_1D$.