
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
Given functions $f(x)=x$ and $g(x)=\ln x$.  
$(1)$ Let $h(x)=af(x)-g(x)$, discuss the extremum of $h(x)$.  
$(2)$ If $x>0$, $af(x)-g(x+1)>0$ always holds, find the range of positive real number $a$.

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