2016 AMC 12B Problem 23

What is the volume of the region in three-dimensional space defined by the inequalities |x|+|y|+|z|\le1 and |x|+|y|+|z-1|\le1?

\textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{4}\qquad\textbf{(C)}\ \frac{1}{3}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ 1

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.