
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
In the complex plane, nonzero complex numbers \( z_{1} \) and \( z_{2} \) lie on a circle centered at \( \mathrm{i} \) (the corresponding point) with radius 1. The real part of \( \bar{z}_{1} \cdot z \) is zero, and the principal value of the argument of \( z_{1} \) is \( \frac{\pi}{6} \). Then \( z_{2} = \) ______.

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