
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 a complete graph of order \( n (\geq 3) \) with edges colored in three colors, and where each color is used at least once. No matter the initial coloring, we can always select \( k \) edges and change the colors of these \( k \) edges to one of the given three colors so that the graph remains connected by edges of that given color. Find the minimum value of \( k \).

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