
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 $2n$ people sitting in a row, and $r$ people $(r < n)$ need to be selected from them. The condition is that no two selected people are sitting in adjacent seats. How many different ways are there to make this selection? Additionally, if the $2n$ people were originally sitting in a circle, how would the situation change?

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