
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  $a>1$  be a constant. In the  $xy$ -plane, let  $A(a,\ 0),\ B(a,\ \ln a)$  and  $C$  be the intersection point of the curve  $y=\ln x$  and the  $x$ -axis. Denote by  $S_1$  the area of the part bounded by the  $x$ -axis, the segment  $BA$  and the curve  $y=\ln x$ (1) For  $1\leq b\leq a$ , let  $D(b,\ \ln b)$ . Find the value of  $b$  such that the area of quadrilateral  $ABDC$  is the closest to  $S_1$  and find the area  $S_2$ .

(2) Find  $\lim_{a\rightarrow \infty} \frac{S_2}{S_1}$ .

*1992 Tokyo University entrance exam/Science*

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