
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
For positive real numbers  $ a,\ b$ , two graphs of the function :  $ x^a$  and  $ \ln bx$  have a tangency of point.

(1) Let  $ (s,\ t)$  be the tangency of point, express this in terms of  $ a$ , then express  $ b$  as the function of  $ a$ .

(2) For  $ h$  with  $ 0 < h < s$ ,  denote the area as  $ A(h)$  of the domain bounded by the line  $ x \equal{} h$  and two curves  $ y \equal{} x^a,\ y \equal{} \ln bx$ .

Express  $ \lim_{h\rightarrow 0} A(h)$  in terms of  $ a$ .

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