
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
Let  $n$  be an integer greater than  $1$ . A certain school has  $1+2+\dots+n$  students and  $n$  classrooms, with capacities for  $1, 2, \dots, n$  people, respectively. The kids play a game in  $k$  rounds as follows: in each round, when the bell rings, the students distribute themselves among the classrooms in such a way that they don't exceed the room capacities, and if two students shared a classroom in a previous round, they cannot do it anymore in the current round. For each  $n$ , determine the greatest possible value of  $k$ .

*Proposed by Victor Domínguez*

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