
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
Find the smallest positive integer  $k$  for which there exists a colouring of the positive integers  $\mathbb{Z}_{>0}$  with  $k$  colours and a function  $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$  with the following two properties:

  $(i)$  For all positive integers  $m,n$  of the same colour,  $f(m+n)=f(m)+f(n).$   $(ii)$  There are positive integers  $m,n$  such that  $f(m+n)\ne f(m)+f(n).$ *In a colouring of  $\mathbb{Z}_{>0}$  with  $k$  colours, every integer is coloured in exactly one of the  $k$  colours. In both  $(i)$  and  $(ii)$  the positive integers  $m,n$  are not necessarily distinct.*

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