
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 qualification football round there are six teams and each two play one versus another exactly once. No two matches are played at the same time. At every moment the difference between the number of already played matches for any two teams is  $0$  or  $1$ . A win is worth  $3$  points, a draw is worth  $1$  point and a loss is worth  $0$  points. Determine the smallest positive integer  $n$  for which it is possible that after the  $n$ -th match all teams have a different number of points and each team has a non-zero number of points.

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