
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 \( n \) and \( k \) be positive integers. There are \( n \) circles in the plane such that every two circles intersect at two distinct points, and all the points of intersection are distinct. Each intersection point must be colored with one of \( n \) distinct colors so that each color is used at least once, and exactly \( k \) distinct colors occur on each circle. Find all values of \( n \geq 2 \) and \( k \) for which such a coloring is possible.

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