
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
Find the smallest possible positive integer n with the following property: 
For all positive integers  $x, y$  and  $z$  with  $x | y^3$  and  $y | z^3$  and  $z | x^3$  always to be true that  $xyz| (x + y + z) ^n$ .

(Gerhard J. Woeginger)

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