
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
In a company of \( n > 1 \) people, some members are acquainted with each other, and some are not. If person \( X \) knows person \( Y \), then \( Y \) also knows \( X \). A person is not considered to be acquainted with themselves. Identify all \( n \) for which there will always be two individuals \( A \) and \( B \) in the company who have the same number of acquaintances within the company, such that there exists either a person \( C \) who is acquainted simultaneously with \( A \) and \( B \), or a person \( D \) who is not acquainted simultaneously with \( A \) and \( B \).

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