
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 9 points in space, no four of which are coplanar, a line segment is drawn between every pair of points. These line segments can be colored blue or red or left uncolored. Find the minimum value of \( n \) such that, for any coloring of any \( n \) of these line segments with either red or blue, the set of these \( n \) line segments necessarily contains a triangle with all edges the same color.

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