
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 a function $g(x)=ax^2-2ax+1+b$ (where $a>0$),
1. If the maximum value is 4 and the minimum value is 1 on the interval [2, 3], find the values of the real numbers $a$ and $b$.
2. If $b=1$, and $g(x)\geq 0$ holds for all $x\in [1, 2)$, find the range of $a$.
3. If $b=1$, and $g(x)\geq 0$ holds for all $a\in [2, 3]$, find the range of $x$.
4. Under the conditions of part (1), let $f(x)=g(|x|)$. If the inequality $f(\log_2 k)>f(2)$ is true, determine the range of the real number $k$.

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