
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 two positive integers  $n$  and  $k$ , we say that  $k$  is *$n$ -ergetic* if:
However the elements of  $M=\{1,2,\ldots, k\}$  are coloured in red and green, there exist  $n$  not necessarily distinct integers of the same colour whose sum is again an element of  $M$  of the same colour. For each positive integer  $n$ , determine the least  $n$ -ergetic integer, if it exists.

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