
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 $z= \frac{1}{2} + \frac{\sqrt{3}}{2}i$ (where $i$ is the imaginary unit). Calculate $z + 2z^{2} + 3z^{3} + 4z^{4} + 5z^{5} + 6z^{6}$.

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