
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 $g(x) = x^2 - 2ax + 1$ has its range as $[0, 4]$ over the interval $[1, 3]$.

(1) Find the value of $a$;  
(2) If the inequality $g(2^x) - k \cdot 4^x \geq 0$ holds for all $x \in [1, +\infty)$, find the range of the real number $k$;  
(3) If the function $y= \frac{g(|2^{x}-1|)}{|2^{x}-1|} + k \cdot \frac{2}{|2^{x}-1|} - 3k$ has three zeros, find the range of the real number $k$.

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