
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
We have  $1000$  balls in  $40$  different colours,  $25$  balls of each colour. Determine the smallest  $n$  for which the following holds: if you place the  $1000$  balls in a circle, in any arbitrary way, then there are always  $n$  adjacent balls which have at least  $20$  different colours.

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