
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
A number of  $n > m \geq 1$  soccer teams play a full tournament, each team meeting (once) each other. Points are awarded:  $2$  for a victory,  $1$  for a tie and  $0$  for a loss. At the end, each team has won half of its points against the  $m$  teams placed last (including each of these teams, who won half of its points against the other  $m-1$ ).

Find all possible values for  $n$  and  $m$ , supported with examples of such tournaments.

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