
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
Consider a rectangle $ABCD$ with dimensions $AB=3$, $BC=8$. The rectangle undergoes the following transformations:
1. A $90^\circ$ clockwise rotation about vertex $B$,
2. A $90^\circ$ clockwise rotation about the new position $C'$ that vertex $C$ moved to after the first rotation,
3. Finally, a $90^\circ$ clockwise rotation around the new position $D''$ that vertex $D$ moved to after the second rotation.
Determine the length of the path traveled by point $A$ throughout these transformations.
A) $\frac{1}{2}\sqrt{73}\pi$
B) $4\pi$
C) $\left(\frac{11}{2} + \sqrt{73}\right)\pi$
D) $\left(\frac{1}{2}\sqrt{73} + \frac{11}{2}\right)\pi$

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