
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 $f(x)=ax+\frac{6}{x}-3$, and the solution set of the inequality $xf\left(x\right) \lt 4$ is $\{x\left|\right.1 \lt x \lt b\}$<br/>$(1)$ Solve the inequality in terms of $x$: $ax^{2}-\left(ac+b\right)x+bc \lt 0\left(c\in R\right)$<br/>$(2)$ Given $g\left(x\right)=mx+7-3m$, if for any $x_{1}\in \left[2,3\right]$, there exists $x_{2}\in \left(1,4\right]$, such that $f({{x_1}})=\frac{{g({{x_2}})}}{{{x_1}}}$ holds, find the range of real number $m$.

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