
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 that the vertices of $\triangle ABC$ correspond to the complex numbers $z_1, z_2, z_3$, and satisfy $\left|z_1\right| = \left|z_2\right| = \left|z_3\right| = 1$, express the orthocenter and 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}
user