
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 arrived at a hotel. They know that the hotel has single rooms numbered 1, 2, ..., $n$, of which $k$ are under renovation (but it is unknown which ones), and the rest are free. Tourists can agree on their actions in advance, after which they go to check in one by one: each checks rooms in any order, finds the first free room not under renovation, and stays there for the night. However, tourists do not want to disturb each other: it is not allowed to check a room where someone else has already checked in. For each $k$, specify the smallest $n$ such that the tourists can all check in without disturbing each other.

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