
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
Given the complex sequence $\left\{z_{n}\right\}$ satisfies: $z_{1}=\frac{\sqrt{3}}{2}$, $z_{n+1}=\overline{z_{n}}\left(1+z_{n}i\right)$ for $n=1, 2, \ldots$, where $\mathrm{i}$ is the imaginary unit. Find the value of $z_{2021}$.

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