
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
Alice is sitting in a teacup ride with infinitely many layers of spinning disks. The largest disk has a radius of 5. Each succeeding disk has its center attached to a point on the circumference of the previous disk and has a radius equal to \( \frac{2}{3} \) of the previous disk. Each disk spins around its center (relative to the disk it is attached to) at a rate of \( \frac{\pi}{6} \) radians per second. Initially, at \( t=0 \), the centers of the disks are aligned on a single line, going outward. Alice is sitting at the limit point of all these disks. After 12 seconds, what is the length of the trajectory that Alice has traced out?

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