
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
Determine the greatest positive integer  $k$  that satisfies the following property: The set of positive integers can be partitioned into  $k$  subsets  $A_1, A_2, \ldots, A_k$  such that for all integers  $n \geq 15$  and all  $i \in \{1, 2, \ldots, k\}$  there exist two distinct elements of  $A_i$  whose sum is  $n.$ *Proposed by Igor Voronovich, Belarus*

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