
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
Circle \(ABC\) consists of a full circle with center \(B\). The region is "rolled" along a flat surface and then up a slope which starts after the circle has been rotated 270 degrees back to its original orientation. If the radius \(BC = \frac{4}{\pi}\) cm, and the slope starts at a 30-degree incline, what is the total length of the path that point \(B\) travels along the flat surface and up the slope by the time \(B\) returns to its original orientation? Assume no slipping occurs.

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