
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
There are  $n$   $(n \ge 3)$  players in a table tennis tournament, in which any two players have a match. Player  $A$  is called not out-performed by player  $B$ , if at least one of player  $A$ 's losers is not a  $B$ 's loser.

Determine, with proof, all possible values of  $n$ , such that the following case could happen: after finishing all the matches, every player is not out-performed by any other player.

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