
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
Let  $T$  denote the  $15$ -element set  $\{10a+b:a,b\in\mathbb{Z},1\le a<b\le 6\}$ . Let  $S$  be a subset of  $T$  in which all six digits  $1,2,\ldots ,6$  appear and in which no three elements together use all these six digits. Determine the largest possible size of  $S$ .

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