
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\$ to \$n\$, of which \$k\$ are under renovation (but it is not known which ones), and the rest are free. 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 free room that is not under renovation, and stays there overnight. However, tourists do not want to disturb each other: they cannot check into a room that someone has already occupied. For each \$k\$, specify the smallest \$n\$ such that the tourists can definitely check-in without disturbing each other.

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