
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  $n$  be an integer greater than  $1$ . For a positive integer  $m$ , let  $S_{m}= \{ 1,2,\ldots, mn\}$ . Suppose that there exists a  $2n$ -element set  $T$  such that
(a) each element of  $T$  is an  $m$ -element subset of  $S_{m}$ ;
(b) each pair of elements of  $T$  shares at most one common element;
and
(c) each element of  $S_{m}$  is contained in exactly two elements of  $T$ .

Determine the maximum possible value of  $m$  in terms of  $n$ .

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