
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
Given points \( P_1, P_2, \ldots, P_n \) on a line, we construct a circle with diameter \( P_i P_j \) for each pair \( i, j \), and we color each circle with one of \( k \) colors. For each \( k \), find all \( n \) for which we can always find two circles of the same color with a common external tangent.

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