
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 $f(x) = \dfrac{g(x)}{x}$.

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

(2) If the inequality $f(2^x) - k \cdot 2^x \geqslant 0$ 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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