
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

The exam consists of \(N \geqslant 3000\) questions. Each of the 31 students learned exactly 3000 of these questions, and any question is known by at least 29 students. Before the exam, the teacher placed all the question cards in a circle. He instructed the students to point to one of the questions and explained that he would give this question to the first student in alphabetical order, the next question clockwise to the second student, the next one to the third student, and so on (each student receives one question).

However, the students could not point to such a card such that each of them would receive a question they knew. What is the smallest \(N\) for which this could happen?

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