
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  $n$  be a positive integer. Each of the numbers  $1,2,3,\ldots,100$  is painted with one of  $n$  colors in such a way that two distinct numbers with a sum divisible by  $4$  are painted with different colors. Determine the smallest value of  $n$  for which such a situation is possible.

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