
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
Let  $n\geq 3$  be an integer.  $n$  balls are arranged in a circle, and are labeled with numbers  $1,2,\cdots,n$  by their order. There are  $10$  colors, and each ball is colored with one of them. Two adjacent balls are colored with different colors. For balls with each color, the total sum of their labeled numbers is independent of color. Find the minimum possible value of  $n$ . 

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