
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 group of  $n$  people play a board game with the following rules:
1) In each round of the game exactly  $3$  people play
2) The game ends after exactly  $n$  rounds
3) Every pair of players has played together at least at one round
Find the largest possible value of  $n$ 

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