
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
Kevin writes a nonempty subset of  $S = \{ 1, 2, \dots 41 \}$  on a board. Each day, Evan takes the set last written on the board and decreases each integer in it by  $1.$  He calls the result  $R.$  If  $R$  does not contain  $0$  he writes  $R$  on the board. If  $R$  contains  $0$  he writes the set containing all elements of  $S$  not in  $R$ . On Evan's  $n$ th day, he sees that he has written Kevin's original subset for the  $1$ st time. Find the sum of all possible  $n.$ 

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