
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 function $f(x)=x^{2}-2a\ln x$ $(a\in\mathbb{R})$, $g(x)=2ax$.  
$(1)$ Find the extreme values of the function $f(x)$.  
$(2)$ If $a > 0$, the function $h(x)=f(x)-g(x)$ has exactly one zero point, find the value of the real number $a$.  
$(3)$ If $0 < a < 1$, for any two distinct real numbers $x_{1}$, $x_{2}$ in the interval $[1,2]$, the inequality $|f(x_{1})-f(x_{2})| > |g(x_{1})-g(x_{2})|$ holds, find the range of values for $a$.

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