
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
On a plane, there is a complete graph of order \( n \) (\( n \geq 3 \)), with its edges colored using three different colors. Suppose it is always possible to select \( k \) edges, recolor these \( k \) edges with one of the three given colors, and ensure that the graph remains connected using that color. No matter how the initial coloring is done, determine the minimum value of \( k \).

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