
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
A natural number \( k \) is given. There are \( n \) poles standing at equal intervals along a road. Misha painted them in \( k \) colors and calculated the distance between each pair of same-colored poles such that no other pole of the same color is between them. All these distances turned out to be different. What is the greatest \( n \) for which this could be the case?

(M. Tikhomirov, F. Petrov)

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