
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
From the set \(\{1, 2, \ldots, n\}\), select triplets \((a, b, c)\) with \(a < b < c\) such that for any two selected triplets \((a, b, c)\) and \((a', b', c')\), at most one of the equalities \(a = a'\), \(b = b'\), or \(c = c'\) holds. What is the maximum number of such triplets that can be selected?

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