
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 the functions $f(x)=\ln x$, $g(x)= \frac {1}{2}ax^{2}-bx$, and let $h(x)=f(x)-g(x)$.  
$(1)$ Find the extremum of the function $F(x)=f(x)-x$;  
$(2)$ If $g(2)=2$ and $a < 0$, discuss the monotonicity of the function $h(x)$;  
$(3)$ If the function $g(x)$ is a linear function of $x$, and the function $h(x)$ has two distinct zeros $x_{1}$, $x_{2}$, find the range of values for $b$.

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