
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, let $z$, $z^2$, and $z + z^2$ be vertices of a triangle. Assume $|z| = 1$ (i.e., $z$ lies on the unit circle). Compute the area of the triangle formed by these vertices.

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