
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 $f(x)=x- \frac {b}{x}$ and $g(x)=2a\ln x$.  
$(1)$ If $b=0$, the graph of the function $f(x)$ is tangent to the graph of the function $g(x)$, find the value of $a$;  
$(2)$ If $a > 0$, $b=-1$, the function $F(x)=xf(x)+g(x)$ satisfies for any $x_{1}$, $x_{2}\in(0,1]$ ($x_{1}\neq x_{2}$), $|F(x_{1})-F(x_{2})| < 3| \frac {1}{x_{1}}- \frac {1}{x_{2}}|$ always holds, find the range of values for $a$;  
$(3)$ If $b=1$, the function $G(x)=f(x)+g(x)$, and $G(x)$ has two extreme points $x_{1}$, $x_{2}$, where $x_{1}\in(0, \frac {1}{3}]$, find the minimum value of $G(x_{1})-G(x_{2})$.

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