
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 $x,$ $y,$ and $z$ be three positive real numbers whose sum is 1. If none of these numbers is more than twice any other, but opposite to our prior scenario, where $z$ was up to twice $x$, assume now that $z$ is exactly twice $y$. Find the minimum value of the product $xyz.$

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