
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
Set  $T\subset\{1,2,\dots,n\}^3$  has the property that for any two triplets  $(a,b,c)$  and  $(x,y,z)$  in  $T$ , we have  $a<b<c$ , and also, we know that at most one of the equalities  $a=x$ ,  $b=y$ ,  $c=z$  holds. Maximize  $|T|$ .

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