
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 championship with 20 teams, where each team plays with every other team exactly once, what is the minimum number of games that must be played to ensure that among any three teams, there will be two teams that have already played against each other?

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