
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 $g(x)=ax-\frac{a}{x}-5\ln x$, where $a\in R$.
(1) Find the range of positive real number $a$ such that $g(x)$ is an increasing function in its domain;
(2) Given the function $h(x)=x^{2}-mx+4$, when $a=2$, if $\exists x_{1}\in(0,1)$, $\forall x_{2}\in[1,2]$, such that $g(x_{1})\geqslant h(x_{2})$ holds, find the range of real number $m$.

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