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

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