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

$(1)$ Find the value of $a$;

$(2)$ If the inequality $g(2^{x})-k\cdot 4^{x}\geqslant 0$ always holds for $x\in [1,+\infty)$, 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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