
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, 15 teams participated, each of which played exactly once with all the others. A win awarded 3 points, a draw 1 point, and a loss 0 points.

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

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