
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 on the interval $[2, 3]$, let $f(x) = \frac{g(x)}{x}$.
(Ⅰ) Determine the values of $a$ and $b$;
(Ⅱ) If the inequality $f(2^x) - k \cdot 2^x \ge 0$ always holds for $x \in [-1, 1]$, find the range of the real number $k$.

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