
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
Danny wants to give his dog, Rex, the maximum space to roam on his circular playground. He plans to tether Rex to a circular fence that has a radius of 20 feet using a 30-foot rope. He is considering two arrangements:
1. Tether Rex to a point halfway along the fence's perimeter.
2. Tether Rex to a point 12 feet from one point directly on the fence's perimeter.
Calculate the difference in area that Rex has to roam in each arrangement.

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