
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, where no four points are coplanar, and each pair of points is connected by a line segment, which can be colored blue, red, or left uncolored. Determine the minimum value of \( n \) such that any way of coloring \( n \) of these line segments either red or blue necessarily contains at least one monochromatic triangle (a triangle whose edges are all the same color).

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