
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
\[
\frac{x^{3}}{x^{3}+2 y^{2} z}+\frac{y^{3}}{y^{3}+2 z^{2} x}+\frac{z^{3}}{z^{3}+2 x^{2} y} \geqslant 1
\]

On the table, there is a pile of \( n \) stones. In one move, any existing pile can be split into two smaller piles. At the same time, the sizes of any two piles on the table must be "similar."

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