
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
12 soccer teams participate in a round-robin tournament. Each pair of teams plays a match, with the winning team earning 2 points and the losing team earning 0 points. If the match results in a draw, each team earns 1 point. Three referees recorded the total points for all teams and obtained three different results: 3,086; 2,018; and 1,238. They calculated the averages of each pair of these numbers and found that the averages were still different. Then, they calculated the average of the initial three numbers. If one of these seven numbers is correct, what is the number of teams \( n \)?

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