
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 hundred mathematicians attending an international combinatorial conference are being accommodated in a hotel where the rooms are numbered from 1 to 100. The reception plans to assign the mathematicians to rooms according to their arrival order and their corresponding room numbers. However, the first guest forgets the instruction and randomly chooses a room. The receptionist then instructs the subsequent guests to take the room corresponding to their arrival number sequentially and, if that room is occupied, to choose any free room that they like. In how many different ways could the guests occupy the rooms?

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