
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
Yesterday, \( n \geq 4 \) people sat around a round table. Each participant remembers only who their two neighbors were, but not which one sat on their left and which one sat on their right. Today, you would like the same people to sit around the same round table so that each participant has the same two neighbors as yesterday (it is possible that yesterday's left-hand side neighbor is today's right-hand side neighbor). You are allowed to query some of the participants: if anyone is asked, they will answer by pointing at their two neighbors from yesterday.

(a) Determine the minimal number \( f(n) \) of participants you have to query in order to be certain to succeed, if later questions must not depend on the outcome of the previous questions. That is, you have to choose in advance the list of people you are going to query, before effectively asking any question.

(b) Determine the minimal number \( g(n) \) of participants you have to query in order to be certain to succeed, if later questions may depend on the outcome of previous questions. That is, you can wait until you get the first answer to choose whom to ask the second question, and so on.

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