
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
Find the surface area generated by rotating around the axis \( O y \):

1) The arc of the circle \( x^{2} + (y - b)^{2} = R^{2} \) between the points where \( y = y_{1} \) and \( y = y_{2} \).
2) The loop of the curve \( 9 a x^{2} = y (3 a - y)^{2} \).

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