
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 \( ABCD \) lies in a plane with \( AB = CD = 3 \) and \( BC = DA = 8 \). The rectangle is rotated \(90^\circ\) clockwise around \( D \), and then it undergoes another \(90^\circ\) clockwise rotation around the new position of point \( C \) following the first rotation. What is the length of the path traveled by point \( A \)?
A) \( 4\pi \)
B) \( \left(6 + \sqrt{73}\right)\pi \)
C) \( \left(4 + \frac{\sqrt{73}}{2}\right)\pi \)
D) \( 10\pi \)

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