
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
In a plane, there are 10 points such that any three points are not collinear. Each pair of points is connected by a line segment, and each line segment is colored with one of \( k \) colors such that for any \( k \) points among the 10, each pair of the \( k \) points is connected by a line segment of a different color. Find all possible positive integers \( k \) satisfying \( 1 \leqslant k \leqslant 10 \).

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