
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 regular polygon with 67 sides, all segments joining two vertices, including the sides of the polygon, are drawn. We choose $n$ of these segments and assign each one a color from 10 possible colors. Find the minimum value of $n$ that guarantees, regardless of which $n$ segments are chosen and how the colors are assigned, that there will always be a vertex of the polygon that belongs to 7 segments of the same color.

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