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

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

$(2)$ If the inequality $f(2^{x})-k\cdot2^{x} \geqslant 0$ has solutions for $x \in [-1,1]$, find the range of the real number $k$.

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