
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  $A=\{1,2,3,\ldots,40\}$ . Find the least positive integer  $k$  for which it is possible to partition  $A$  into  $k$  disjoint subsets with the property that if  $a,b,c$  (not necessarily distinct) are in the same subset, then  $a\ne b+c$ .

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