
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
Let  $k$  and  $n$  be positive integers, with  $k \geq 2$ . In a straight line there are  $kn$  stones of  $k$  colours, such that there are  $n$  stones of each colour. A *step* consists of exchanging the position of two adjacent stones. Find the smallest positive integer  $m$  such that it is always possible to achieve, with at most  $m$  steps, that the  $n$  stones are together, if:
a)  $n$  is even.
b)  $n$  is odd and  $k=3$ 

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