
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
As shown in the diagram, there are three adjacent circles of different sizes with radii $9 \text{ cm}$, $3 \text{ cm}$, and $1 \text{ cm}$. The middle circle is rolling clockwise downward along the inner side of the larger circle, and the smaller circle is rolling counterclockwise upward along the inner side of the middle circle, both at a speed of 1 cm per second along their circumferences. If a fixed arrow is on the smaller circle, what is the angle of rotation of the arrow (relative to the center of the small circle) when the middle circle completes one full revolution and returns to its starting point? Provide the answer in degrees.

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