
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 $f(x)=x^{2}-ax$, $g(x)=\ln x$, and $h(x)=f(x)+g(x)$.
(1) If the interval in which $h(x)$ is decreasing is $(\frac{1}{2},1)$, find the value of the real number $a$;
(2) If $f(x)\geqslant g(x)$ holds for any $x$ in the domain, find the range of values for the real number $a$;
(3) Suppose $h(x)$ has two extreme points $x_{1}$, $x_{2}$, and $x_{1}\in(0,\frac{1}{2})$. If $h(x_{1})-h(x_{2})>m$ holds, find the maximum value of $m$.

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