
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
Let $z_1$ and $z_2$ be the complex roots of $z^2 + pz + q = 0,$ where $p$ and $q$ are complex numbers. In the complex plane, $0, z_1,$ and $z_2$ form the vertices of an equilateral triangle. Given that one vertex of the triangle represents a $120^\circ$ rotation from another vertex, find $\frac{p^2}{q}.$

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