
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
Arc \( AD \) is a semi-circle with center \( B \). The shaded region \( ABD \) is "rolled" along a straight board \( PQ \) until it returns to its starting orientation, with point \( B \) first touching point \( B' \). If \( BD = 3 \) cm, determine the length of the path that point \( B \) travels. Provide the answer in simplest form.

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