
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
We want to color the edges of a complete graph with \( n \geq 3 \) vertices using \( k \) colors, where an edge can receive more than one color, but 

(1) no single color appears on every edge,

(2) no single edge contains all colors, and

(3) in any triangle, each color appears on an odd number of edges.

What is the minimum value of \( k \)?

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