
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 say that a group of  $k$  boys is  $n-acceptable$  if removing any boy from the group one can always find, in the other  $k-1$  group,  a group of  $n$  boys such that everyone knows each other. For each  $n$ , find the biggest  $k$  such that in any group of  $k$  boys that is  $n-acceptable$  we must always have a group of  $n+1$  boys such that everyone knows each other.

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