
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 positive integers  $n,k$ ,  $n \ge 2$ . Find the minimum constant  $c$  satisfies the following assertion:
For any positive integer  $m$  and a  $kn$ -regular graph  $G$  with  $m$  vertices, one could color the vertices of  $G$  with  $n$  different colors, such that the number of monochrome edges is at most  $cm$ .

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