
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  $300$  participants to a mathematics competition. After the competition some of the contestants play some games of chess. Each two contestants play at most one game against each other. There are no three contestants, such that each of them plays against each other. Determine the maximum value of  $n$  for which it is possible to satisfy the following conditions at the same time: each contestant plays at most  $n$  games of chess, and for each  $m$  with  $1 \le m \le n$ , there is a contestant playing exactly  $m$  games of chess.

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