
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let \[z=\frac{1+i}{\sqrt{2}}.\] Calculate the expression \[\left(z^{1^3}+z^{2^3}+z^{3^3}+\dots+z^{{8}^3}\right) \cdot \left(\frac{1}{z^{1^3}}+\frac{1}{z^{2^3}}+\frac{1}{z^{3^3}}+\dots+\frac{1}{z^{{8}^3}}\right).\]

Present the answer in LaTex format: \boxed{Your answer}
user