
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 \( n \).

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