
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
We extend the original problem to require that every nine-element subset of the set has the triangle property. Consider sets $\{a, a+1, a+2, \ldots, n\}$ of consecutive positive integers, where $a > 3$, all of whose nine-element subsets have the triangle property. What is the largest possible value of $n$ if $a = 6$?

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