
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
Rectangle $ABCD$ has sides $AB = CD = 4$ and $BC = DA = 8$. The rectangle is rotated $90^\circ$ clockwise about vertex $D$, then $90^\circ$ clockwise about the point $C$ as moved after the first rotation, and finally $90^\circ$ clockwise about the point $B$ as moved after the second rotation. What is the length of the path traveled by point $A$?
A) $8\sqrt{5}\pi$
B) $4\pi$
C) $(4 + 4\sqrt{5})\pi$
D) $8\pi$
E) $12\pi$

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