
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
One night, 100 tourists arrive at a hotel. They know that the hotel has single rooms numbered 1, 2, ..., \( n \), of which \( k \) are under repair (but it is not known which ones), and the remaining rooms are available. The tourists can agree in advance on their actions, after which they take turns checking in: each tourist checks the rooms in any order, finds the first available room that is not under repair, and stays there for the night. However, the tourists do not want to disturb each other: checking a room that is already occupied by someone is not allowed. For each \( k \), indicate the smallest \( n \) for which the tourists are guaranteed to be able to check in without disturbing each other.

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