
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
For finite sets  $A,M$  such that  $A \subseteq M \subset \mathbb{Z}^+$ , we define   $$ f_M(A)=\{x\in M \mid x\text{ is divisible by an odd number of elements of }A\}. $$   Given a positive integer  $k$ , we call  $M$  *k-colorable* if it is possible to color the subsets of  $M$  with  $k$  colors so that for any  $A \subseteq M$ , if  $f_M(A)\neq A$  then  $f_M(A)$  and  $A$  have different colors.
Determine the least positive integer  $k$  such that every finite set  $M \subset\mathbb{Z}^+$  is k-colorable.

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