
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 functions $f(x)=\ln x-\dfrac{a}{x}$ and $g(x)=f(x)+ax-6\ln x$, where $a\in R$ is a constant.
1. When $a=1$, determine the monotonicity of $f(x)$.
2. If $g(x)$ is an increasing function in its domain, find the range of real numbers for $a$.
3. Let $h(x)=x^{2}-mx+4$. When $a=2$, if there exists $x_{1}\in \left[1,2\right]$ such that $g(x_{1})\geqslant h(x_{2})$ holds, find the range of real numbers for $m$.

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