
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
There is a complete graph of order $n (\geq 3)$ on a plane, where the edges are colored using three colors, with each color used at least once. No matter how the initial coloring is done, we can always select $k$ edges from the complete graph, and by changing the colors of these $k$ edges to one of the given three colors, the graph can be made 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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