
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
120 schools each send 20 people to form 20 teams, with each team having exactly 1 person from each school. Find the smallest positive integer \( k \) such that when \( k \) people are selected from each team, there will be at least 20 people from the same school among all the selected individuals.

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