
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
The vertices of triangle \( \triangle ABC \) correspond to complex numbers \( z_{1}, z_{2}, z_{3} \), and \( \left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=1 \). Express the center of the nine-point circle of \( \triangle ABC \) in terms of \( z_{1}, z_{2}, z_{3} \).

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