
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 the World and European championships. Among them, there are $k$ European teams, and the results of the matches between these teams in the World Championship are counted for the European Championship. The championship is held in a round-robin format.

What is the largest value of $k$ such that the team scoring the strictly highest number of points in the European Championship will score the strictly lowest number of points in the World Championship, if this is:

a) a hockey championship (draws are allowed)?

b) a volleyball championship (draws are not allowed)?

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