
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
If the distinct non-zero numbers $x ( y - z),~ y(z - x),~ z(x - y )$ form a geometric progression with terms related as $a, ar, ar^3$, find the equation satisfied by $r$.
- **A)** $r^2 + r + 1 = 0$
- **B)** $r^2 - r + 1 = 0$
- **C)** $r^4 + r^2 - 1 = 0$
- **D)** $r^3 + r + 1 = 0$
- **E)** $r^3 - r^2 + 1 = 0$

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