
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 nine points in space such that no four of them lie on the same plane. Each pair of points is connected by an edge (i.e., a line segment), and each such edge is either colored red or blue or left uncolored. Determine the smallest value of \( n \) such that whenever the number of colored edges is exactly \( n \), it is always true that the set of colored edges necessarily contains a triangle in which all edges have the same color.

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