
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
Let  $S$  be the domain surrounded by the two curves  $C_1:y=ax^2,\ C_2:y=-ax^2+2abx$   for constant positive numbers  $a,b$ .
Let  $V_x$  be the volume of the solid formed by the revolution of  $S$  about the axis of  $x$ ,  $V_y$  be the volume of the solid formed by the revolution of  $S$  

about the axis of   $y$ . Find the ratio of   $\frac{V_x}{V_y}$ .

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