
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
Consider the 60 vertices of a regular 60-gon. Among these vertices, 10 are colored red, 20 are colored yellow, and the remaining 30 are colored blue. Find the smallest positive integer $m$ such that it is always possible to find $m$ straight lines that do not pass through any vertices, dividing the plane into regions where each region contains vertices of only one color.


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