
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 $g(x)=ax^{2}-2ax+b$ where $b>0$, the maximum value is $1$ and the minimum value is $0$ when $x\in [1,2]$. Let $f(x)=\frac{{g(x)}}{x}$.
1. Find the values of real numbers $a$ and $b$.
2. If the inequality $g(2^{x})-k\cdot 4^{x}+1\geqslant 0$ holds for $x\in [-1,1]$, find the range of real number $k$.
3. If the equation $f({|{{{log}_2}x}|})+\frac{{2m}}{{|{{{log}_2}x}|}}-3m-1=0$ has four distinct real number solutions, find the range of real number $m$.

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