
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
Each pair of vertices of a regular  $67$ -gon is joined by a line segment. Suppose  $n$  of these segments are selected, and each of them is painted one of ten available colors. Find the minimum possible value of  $n$  for which, regardless of which  $n$  segments were selected and how they were painted, there will always be a vertex of the polygon that belongs to seven segments of the same color.

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