
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
Considering sets $\{3, 4, 5, \ldots, n\}$ of consecutive positive integers, find the largest value of $n$ such that every fifteen-element subset of it has the triangle property, where a subset has the triangle property if it includes three elements that can form the sides of a triangle with a non-zero area.

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