AMC 12 2019a Problem 21

Let $$z=\frac{1+i}{\sqrt{2}}.$$What is $$\left(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}\right) \cdot \left(\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}}\right)?$$ $\textbf{(A) } 18 \textbf{(B) } 72-36\sqrt2 \textbf{(C) } 36 \textbf{(D) } 72 \textbf{(E) } 72+36\sqrt2$