
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 method of factorization of the following expressions:<br/>①$x^{2}-1=\left(x-1\right)\left(x+1\right)$<br/>②$x^{3}-1=x^{3}-x+x-1(Step 1)$<br/>$x(x^{2}-1)+x-1(Step 2)$<br/>$=x\left(x-1\right)\left(x+1\right)+\left(x-1\right)(Step 3)$<br/>$=\left(x-1\right)\left[x\left(x+1\right)+1\right](Step 4)$<br/>$=\left(x-1\right)(x^{2}+x+1)(Step 5)$<br/>③$x^{4}-1=x^{4}-x+x-1$<br/>$=x(x^{3}-1)+x-1$<br/>$=x\left(x-1\right)(x^{2}+x+1)+\left(x-1\right)$<br/>$=\left(x-1\right)[x(x^{2}+x+1)+1]$<br/>$=\left(x-1\right)(x^{3}+x^{2}+x+1)$<br/>$(1)$ In ②, the method of factorization used from Step 3 to Step 4 is ______；<br/>$(2)$ Imitate the above method and try to factorize $x^{5}-1$；<br/>$(3)$ Observe the above results and directly write out the factorization of $x^{n}-1$；<br/>$(4)$ According to the above conclusion, try to find the value of $2^{5}+2^{4}+2^{3}+2^{2}+2+1$.

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