
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
Determine the smallest possible positive integer \( n \) with the following property: For all positive integers \( x, y, \) and \( z \) such that \( x \mid y^{3} \), \( y \mid z^{3} \), and \( z \mid x^{3} \), it always holds that \( x y z \mid (x+y+z)^{n} \).

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