
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
A train wheel with a diameter of 1 meter is rolling on a horizontal track. A connecting rod is attached halfway down the length of a spoke. This connection point traces a curve known as an extended cycloid. What is the radius of the circles that best approximate the curve in the vicinity of the highest and lowest points on the path of this point?

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