
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 football tournament, there were 15 teams, and each team played with every other team exactly once. A victory awarded 3 points, a draw awarded 1 point, and a loss awarded 0 points.

After the tournament ended, it turned out that some 6 teams each scored at least $N$ points. What is the maximum integer value that $N$ can take?

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