
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
In a round robin chess tournament each player plays every other player exactly once. The winner of each game gets  $ 1$  point and the loser gets  $ 0$  points. If the game is tied, each player gets  $ 0.5$  points. Given a positive integer  $ m$ , a tournament is said to have property  $ P(m)$  if the following holds: among every set  $ S$  of  $ m$  players, there is one player who won all her games against the other  $ m\minus{}1$  players in  $ S$  and one player who lost all her games against the other  $ m \minus{} 1$  players in  $ S$ . For a given integer  $ m \ge 4$ , determine the minimum value of  $ n$  (as a function of  $ m$ ) such that the following holds: in every  $ n$ -player round robin chess tournament with property  $ P(m)$ , the final scores of the  $ n$  players are all distinct.

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