
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
Read the following material: The overall idea is a common thinking method in mathematical problem solving: Here is a process of a student factorizing the polynomial $(x^{2}+2x)(x^{2}+2x+2)+1$. Regard "$x^{2}+2x$" as a whole, let $x^{2}+2x=y$, then the original expression $=y^{2}+2y+1=\left(y+1\right)^{2}$, and then restore "$y$".

**Question:**  
(1) ① The student's factorization result is incorrect, please write down the correct result directly ______;  
② According to material $1$, please try to imitate the above method to factorize the polynomial $(x^{2}-6x+8)(x^{2}-6x+10)+1$;  
(2) According to material $1$, please try to imitate the above method to calculate:  
$(1-2-3-\ldots -2020)\times \left(2+3+\ldots +2021\right)-\left(1-2-3-\ldots -2021\right)\times \left(2+3+\ldots +2020\right)$.

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