
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  $a$  be a constant. In the  $xy$  palne, the curve  $C_1:y=\frac{\ln x}{x}$  touches  $C_2:y=ax^2$ .
Find the volume of the solid generated by a rotation of the part enclosed by  $C_1,\ C_2$  and the  $x$  axis about the  $x$  axis.

*2011 Yokohama National Universty entrance exam/Engineering*

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