Square, Triangles, Square

If the area of square $ABDC$ is 1, what is the area of the shaded region? Don't forget to read the details and descriptions.

Details and descriptions:

• $AX = \frac{1}{3} AB$ , $BY = \frac 13 BD$ , $DZ = \frac 13 CD$ , and $CW = \frac 13 AC$
• $AY = BZ = DW = CX$
• We do not know some of the angles yet.