
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 $z=\frac{1-\sqrt{3}i}{2}$. Consider the expression $\left(z + z^{3} + z^{5} + \dots + z^{15}\right) \cdot \left(\frac{1}{z} + \frac{1}{z^{3}} + \frac{1}{z^{5}} + \dots + \frac{1}{z^{15}}\right)$. Calculate the product of this expression.

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