
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
Celeste has an unlimited amount of each type of  $n$  types of candy, numerated type 1, type 2, ... type n. Initially she takes  $m>0$  candy pieces and places them in a row on a table. Then, she chooses one of the following operations (if available) and executes it: $1.$  She eats a candy of type  $k$ , and in its position in the row she places one candy type  $k-1$  followed by one candy type  $k+1$  (we consider type  $n+1$  to be type 1, and type 0 to be type  $n$ ). $2.$  She chooses two consecutive candies which are the same type, and eats them.

Find all positive integers  $n$  for which Celeste can leave the table empty for any value of  $m$  and any configuration of candies on the table. $\textit{Proposed by Federico Bach and Santiago Rodriguez, Colombia}$ 

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