
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 $g(x)=ax^{2}-2ax+1+b$ $(a\neq 0,b < 1)$ has its maximum value of $4$ and minimum value of $0$ in the interval $[0,3]$. Let $f(x)=\frac{g(x)}{x}$.

$(1)$ Find the values of $a$ and $b$;

$(2)$ The inequality $f(2^{x})-k\cdot2^{x}\geqslant 0$ holds for all $x\in[-2,-1]$, find the range of the real number $k$;

$(3)$ If the equation $f\left( |2^{x}-1| \right)+k\left( \frac{2}{|2^{x}-1|}-3 \right)=0$ has three distinct real roots, find the range of the real number $k$.

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