
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  $n{}$  be a positive integer, and let  $\mathcal{C}$  be a collection of subsets of  $\{1,2,\ldots,2^n\}$  satisfying both of the following conditions:[list=1]
[*]Every  $(2^n-1)$ -element subset of  $\{1,2,\ldots,2^n\}$  is a member of  $\mathcal{C}$ , and
[*]Every non-empty member  $C$  of  $\mathcal{C}$  contains an element  $c$  such that  $C\setminus\{c\}$  is again a member of  $\mathcal{C}$ .
[/list]Determine the smallest size  $\mathcal{C}$  may have.

*Serbia, Pavle Martinovic ́*

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