
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
On the coordinate plane, Let  $C$  be the graph of  $y=(\ln x)^2\ (x>0)$  and for  $\alpha >0$ , denote  $L(\alpha)$  be  the tangent line of  $C$  at the point  $(\alpha ,\ (\ln \alpha)^2).$ (1) Draw the graph.

(2) Let  $n(\alpha)$  be the number of the intersection points of  $C$  and  $L(\alpha)$ . Find  $n(\alpha)$ .

(3) For  $0<\alpha <1$ , let  $S(\alpha)$  be the area of the region bounded by  $C,\ L(\alpha)$  and the  $x$ -axis. Find  $S(\alpha)$ .

2010 Tokyo Institute of Technology entrance exam, Second Exam.

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