
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
During a secret meeting, 20 trainees elect their favorite supervisor. Each trainee votes for two supervisors. It is known that for any two trainees, there is always at least one supervisor for whom both have voted. What is the minimum number of votes received by the supervisor who wins the election?

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