Task 662

The task

A solid right prism $ABCDEF$ has a height of $16,$ as shown. Also, its bases are equilateral triangles with side length $12.$ Points $X,$ $Y,$ and $Z$ are the midpoints of edges $AC,$ $BC,$ and $DC,$ respectively. A part of the prism above is sliced off with a straight cut through points $X,$ $Y,$ and $Z.$ Determine the surface area of solid $CXYZ,$ the part that was sliced off. [asy] pair A, B, C, D, E, F, X, Y, Z; A=(0,0); B=(12,0); C=(6,-6); D=(6,-22); E=(0,-16); F=(12,-16); X=(A+C)/2; Y=(B+C)/2; Z=(C+D)/2; draw(A--B--C--A--E--D--F--B--C--D); draw(X--Y--Z--X, dashed); label("$A$", A, NW); label("$B$", B, NE); label("$C$", C, N); label("$D$", D, S); label("$E$", E, SW); label("$F$", F, SE); label("$X$", X, SW); label("$Y$", Y, SE); label("$Z$", Z, SE); label("12", (A+B)/2, dir(90)); label("16", (B+F)/2, dir(0)); [/asy]