
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
Points  $A_1, A_2,\ldots, A_n$  are found on a circle in this order. Each point  $A_i$  has exactly  $i$  coins. A move consists in taking two coins from two points (may be the same point) and moving them to adjacent points (one move clockwise and another counter-clockwise). Find all possible values of  $n$  for which it is possible after a finite number of moves  to obtain a configuration with each point  $A_i$  having  $n+1-i$  coins.

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