Slicing a Cube Through a Face Center

In a cube $ABCD\text{-}A_1B_1C_1D_1$ with edge length $4$, $P$ is the center of the square face $BCC_1B_1$, and $Q$ is the midpoint of edge $DD_1$. The plane through the points $A$, $P$, $Q$ divides the cube into two parts. Find the volume of the smaller part.