
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 a pond there are  $n \geq 3$  stones arranged in a circle. A princess wants to label the stones with the numbers  $1, 2, \dots, n$  in some order and then place some toads on the stones. Once all the toads are located, they start jumping clockwise, according to the following rule: when a toad reaches the stone labeled with the number  $k$ , it waits for  $k$  minutes and then jumps to the adjacent stone.

What is the greatest number of toads for which the princess can label the stones and place the toads in such a way that at no time are two toads occupying a stone at the same time?**Note:** A stone is considered occupied by two toads at the same time only if there are two toads that are on the stone for at least one minute.

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