
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
There are 2010 boxes labeled  $B_1,B_2,\dots,B_{2010},$  and  $2010n$  balls have been distributed among them, for some positive integer  $n.$  You may redistribute the balls by a sequence of moves, each of which consists of choosing an  $i$  and moving *exactly*  $i$  balls from box  $B_i$  into any one other box. For which values of  $n$  is it possible to reach the distribution with exactly  $n$  balls in each box, regardless of the initial distribution of balls?

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