
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
Given that \( k \) is a positive integer, a tournament committee arranges a competition with \( 2k \) players where each pair of players compete in one match. Exactly one match is played per day. Each player participates in their first match on the day they arrive and their last match on the day they leave. The committee must pay the hotel one coin per day for each player staying at the hotel. The committee wants to design a schedule to minimize the total amount paid to the hotel for all players' stays. Find this minimum value.

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