
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
Observe the following equations: $\left(x\neq 0\right)$<br/>$(\frac{1}{x}-1)(\frac{1}{x}+1)=\frac{1}{{x}^{2}}-1$,<br/>$(\frac{1}{x}-1)(\frac{1}{{x}^{2}}+\frac{1}{x}+1)=\frac{1}{{x}^{3}}-1$,<br/>$(\frac{1}{x}-1)(\frac{1}{{x}^{3}}+\frac{1}{{x}^{2}}+\frac{1}{x}+1)=\frac{1}{{x}^{4}}-1$.<br/>$(1)$ From the above equations and calculation results, according to the pattern you discovered, directly write the following blank:<br/>$(\frac{1}{x}-1)(\frac{1}{{x}^{7}}+\frac{1}{{x}^{6}}+\frac{1}{{x}^{5}}+\frac{1}{{x}^{4}}+\frac{1}{{x}^{3}}+\frac{1}{{x}^{2}}+\frac{1}{x}+1)=$______；<br/>$(2)$ Using the holistic approach of mathematics, let $\frac{1}{x}=m$, decompose the inner expression: $(m^{7}+m^{6}+m^{5}+m^{4}+m^{3}+m^{2}+m+1)\left(m\neq 1\right)$；<br/>$(3)$ Given $1+2+2^{2}+2^{3}+2^{4}+2^{5}+2^{6}+2^{7}=a\cdot b\cdot c\cdot d$, where $a$, $b$, $c$, $d$ are all positive integers, and $a \gt b \gt c \gt d$. Simplify and find the value of $(-\frac{b}{cd})^{2}\div (-\frac{5b}{17c})\times \frac{6d}{a}$.

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