
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 a positive integer \( k \), a tennis tournament committee arranges \( 2k \) players to participate in the tournament such that each pair of players plays one match against each other, with exactly one match occurring each day. Each player participates in their first match on the day they arrive at the competition venue and their last match on the day they leave. Every day, the committee must pay the hotel one coin for each player staying at the hotel. The committee wants to design a schedule that minimizes the total payment for all players staying at the hotel. Find this minimum total cost.

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