
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
Geoff has an infinite stock of sweets, which come in  $n$  flavours. He arbitrarily distributes some of the sweets amongst  $n$  children (a child can get sweets of any subset of all flavours, including the empty set). Call a distribution  $k-\textit{nice}$  if every group of  $k$  children together has sweets in at least  $k$  flavours. Find all subsets  $S$  of  $\{ 1, 2, \dots, n \}$  such that if a distribution of sweets is  $s$ -nice for all  $s \in S$ , then it is  $s$ -nice for all  $s \in \{ 1, 2, \dots, n \}$ .

*Proposed by Kyle Hess, USA*

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