
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  $M$  be a set of  $2017$  positive integers. For any subset  $A$  of  $M$  we define  $f(A) := \{x\in M\mid \text{ the number of the members of }A\,,\, x \text{ is multiple of, is odd }\}$ . 
Find the minimal natural number  $k$ , satisfying the condition:  for any  $M$ , we can color all the subsets of  $M$  with  $k$  colors, such that whenever  $A\neq f(A)$ ,  $A$  and  $f(A)$  are colored with different colors.

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