Let \[z=\frac{1+i}{\sqrt{2}}.\] Calculate the value of \[\left(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}\right) \cdot \left(\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}}\right).\]

Let's think step by step and output the final answer within \boxed{}.
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