
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

At night, 100 tourists arrive at a hotel. They know that the hotel has single rooms numbered 1,2, …, n, of which k are under renovation (it is not known which ones), and the rest are available. The tourists can agree on their actions in advance, after which they go to check in one by one: each checks the rooms in any order, finds the first available room not under renovation, and stays there for the night. However, the tourists do not want to disturb each other: no one can check a room that someone has already occupied. For each k, determine the smallest n such that the tourists can guarantee they all get a room without disturbing one another.

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