
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
An interstellar hotel has  $100$  rooms with capacities  $101,102,\ldots, 200$  people. These rooms are occupied by  $n$  people in total. Now a VIP guest is about to arrive and the owner wants to provide him with a personal room. On that purpose, the owner wants to choose two rooms  $A$  and  $B$  and move all guests from  $A$  to  $B$   without exceeding its capacity. Determine the largest  $n$  for which the owner can be sure that he can achieve his goal no matter what the initial distribution of the guests is.  

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