
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
A company, during their vacation, managed to conduct a round-robin ping pong tournament such that each day one match was played. Additionally, each day one game was played with all the chess sets found at the resort. On the last day of their holiday, they determined that no two days had the same chess player pairings, but the next day they would only be able to arrange pairings that had already been made during the vacation. (In this context, it didn't matter which color pieces each player used.) How many members were in the company and how many days did the vacation last?

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