
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 the distinct non-zero numbers $x, y, z$ such that $x(y-z)$, $y(z-x)$, and $z(x-y)$ form a geometric progression with common ratio $r$, and additionally, $x + y + z = 0$, find the value of $r$.
A) $\frac{-1 + i\sqrt{3}}{2}$
B) $\frac{-1 - i\sqrt{3}}{2}$
C) $\frac{1 + i\sqrt{3}}{2}$
D) $\frac{1 - i\sqrt{3}}{2}$

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