
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 quarter-circle with center $B$ and radius $\frac{4}{\pi}$ cm is initially in its upright position on a flat board. The quarter-circle rolls to its right, completes a full rotation, and covers the distance necessary for a complete upright reposition. In the middle of its path, the quarter-circle travels as a straight line without rolling for an additional $2$ cm before continuing its rolling motion. Finally, it rolls back to the original upright position along the same path. Determine the total length of the path that point $B$ travels.

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