
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 $f(x) = a \cdot 4^x - a \cdot 2^{x+1} + 1 - b$ ($a > 0$) has a maximum value of 9 and a minimum value of 1 on the interval $[1, 2]$,
(1) Find the values of $a$ and $b$;
(2) If the inequality $f(x) - k \cdot 4^x \geq 0$ has a solution 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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