
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: Factorization: $\left(x+y\right)^{2}+2\left(x+y\right)+1$, Solution: Let $\left(x+y\right)=A$, then the original expression $=A^{2}+2A+1=\left(A+1\right)^{2}$, then restore "$A$", we get the original expression $=\left(x+y+1\right)^{2$. The above problem-solving method uses the "whole idea," which is a commonly used thinking method in mathematical problem-solving. Please answer the following questions:  
$(1)$ Factorization: $1-2\left(x-y\right)+\left(x-y\right)^{2}$;  
$(2)$ For what value of $n$, the algebraic expression $(n^{2}-2n-3)(n^{2}-2n+5)+17$ has the minimum value? What is the minimum value?

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