
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
Initially, there are  $n$  red boxes numbered with the numbers  $1,2,\dots ,n$  and  $n$  white boxes numbered with the numbers  $1,2,\dots ,n$  on the table. At every move, we choose  $2$  different colored boxes and put a ball on each of them. After some moves, every pair of the same numbered boxes has the property of either the number of balls from the red one is  $6$  more than the number of balls from the white one or the number of balls from the white one is  $16$  more than the number of balls from the red one. With that given information find all possible values of  $n$ 

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