
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  $C: y=\ln x$ . For each positive integer  $n$ , denote by  $A_n$  the area of the part enclosed by the line passing through two points  $(n,\ \ln n),\ (n+1,\ \ln (n+1))$  and denote by  $B_n$  that of the part enclosed by the tangent line at the point  $(n,\ \ln n)$ ,  $C$  and the line  $x=n+1$ . Let  $g(x)=\ln (x+1)-\ln x$ .

(1) Express  $A_n,\ B_n$  in terms of  $n,\ g(n)$  respectively.

(2) Find  $\lim_{n\to\infty} n\{1-ng(n)\}$ .

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