
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 hotel, 100 tourists arrive at night. They know that the hotel has single rooms numbered 1, 2, ⋯, n, out of which k are under renovation (but they do not know which ones), and the rest are free. The tourists can agree on their actions in advance, and then they leave to check into their rooms one by one: each tourist checks the rooms in any order, finds the first free room that is not under renovation, and stays there for the night. However, tourists do not want to disturb one another: they cannot check a room where someone is already staying. For each k, specify the smallest n, such that the tourists can be guaranteed to check in without disturbing each other.

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