
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

There are $n$ students seated around a round table for a meal, where $n$ is a given positive integer greater than 1. Each student needs to buy a dinner voucher costing 50 yuan. Let's assume exactly $r$ students (where $1 \leq r \leq n$) have a 100 yuan bill, while the rest have a 50 yuan bill. When the server named $K$ comes to the round table to sell dinner vouchers, $K$ does not have any change. Regardless of how the $n$ students are arranged around the table, $K$ can start from any student and move in either direction (clockwise or counterclockwise) around the table to sell dinner vouchers, ensuring that all students can buy their vouchers without the problem of making change. Find the maximum value of $r$.

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