
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
$n$ players participate in a chess tournament where each player plays a game with every other player. The winner of a game receives 1 point, the loser receives 0 points, and a draw awards each player 0.5 points. A tournament is said to have property $P(m)$ if, among any $m$ players, there is one player who has defeated the other $m-1$ players, and there is one player who has lost to the other $m-1$ players. For a given $m$ ($m \geq 4$), find the minimum value of $n$, denoted by $f(m)$, such that in any tournament with property $P(m)$, all $n$ players have different scores.

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