
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 greatest positive integer  $n$  for which there exist  $n$  nonnegative integers  $x_1, x_2,\ldots , x_n$ , not all zero, such that for any  $\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n$  from the set  $\{-1, 0, 1\}$ , not all zero,  $\varepsilon_1 x_1 + \varepsilon_2 x_2 + \cdots + \varepsilon_n x_n$  is not divisible by  $n^3$ .

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