
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 us have  $n$  (  $n>3$ ) balls with different rays. On each ball it is written an integer number. Determine the greatest natural number  $d$  such that for any numbers written on the balls, we can always find at least 4 different ways to choose some balls with the sum of the numbers written on them divisible by  $d$ .

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