
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
Given a set \( S = \{1, 2, \ldots, n\} \), and subsets \( A_{i} = \{a_{i}, b_{i}, c_{i}\} \) with \( a_{i} < b_{i} < c_{i} \) (for \( i = 1, 2, \ldots, m \)), which are three-element subsets of \( S \), determine the maximum value of \( m \) such that for any \( A_{i} \) and \( A_{j} \) (where \( 1 \leq i < j \leq m \)), at most one of the conditions \( a_{i} = a_{j} \), \( b_{i} = b_{j} \), or \( c_{i} = c_{j} \) holds.

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