
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  $n\ge 3$  be a fixed integer. There are  $m\ge n+1$  beads on a circular necklace. You wish to paint the beads using  $n$  colors, such that among any  $n+1$  consecutive beads every color appears at least once. Find the largest value of  $m$  for which this task is  $\emph{not}$  possible.

*Carl Schildkraut, USA*

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