
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 $f(x)=\ln x$ and $g(x)=\frac{1}{2}ax+b$.  
$(1)$ If the curve of $f(x)$ is tangent to $g(x)$ at $x=2$, find the expression of $g(x)$.  
$(2)$ If $h(x)=\frac{m(x-1)}{x+1}-f(x)$ is a decreasing function in $(0,+\infty)$, find the range of real number $m$.

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