
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 $f(x)=x^{2}-a\ln x$, where $a\in R$.
(1) Discuss the monotonicity of the function $f(x)$;
(2) When $a > 0$, if the minimum value of $f(x)$ is $1$, find the value of $a$;
(3) Let $g(x)=f(x)-2x$, if $g(x)$ has two extreme points $x_{1}$, $x_{2}(x_{1} < x_{2})$ in $\left[\frac{1}{4}, \frac{3}{4}\right]$, find the range of $g(x_{1})-g(x_{2})$.

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