
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 $f(x)=\begin{cases} 4-8\left| x-\dfrac{3}{2} \right|, & 1\leqslant x\leqslant 2, \\ \dfrac{1}{2}f\left(\dfrac{x}{2}\right), & x > 2 \end{cases}$, find the sum of all zeros of the function $g(x)=xf(x)-6$ in the interval $\left[ 1,2^{n} \right]$ $(n\in \mathbb{N})$.

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