
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
Find the largest positive integer $k$ such that the set of positive integers can be divided into $k$ subsets $A_{1}, A_{2}, \cdots, A_{k}$, so that for all integers $n(n \geq 15)$ and for all $i \in \{1, 2, \cdots, k\}$, there exist two distinct elements in $A_{i}$ whose sum is $n$.

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