
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
There are 20 teams participating in both the World Championship and the European Championship. Among these, $k$ are European teams, and the results of their matches in the World Championship count towards the European Championship. The championships are conducted in a round-robin format. If the matches follow:

1. Ice Hockey rules (draws are allowed, winning team gets 2 points, losing team gets 0 points, and each team gets 1 point for a draw),
2. Volleyball rules (no draws, winning team gets 1 point, losing team gets 0 points),

What is the maximum value of $k$ such that a team that scores the most points in the European Championship can score the fewest points in the World Championship?

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