
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 an intergalactic hotel, there are 100 rooms with capacities of $101, 102, \ldots, 200$ people. A total of $n$ people are staying in these rooms. A VIP guest arrives, and an entire room needs to be vacated for them. The hotel manager chooses one room and relocates all its occupants to another single room. What is the maximum value of $n$ such that the manager can always free up a room, regardless of the current distribution of guests?

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