
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 materials:<br/>Material $1$: When factoring a quadratic trinomial in the form of $x^{2}+px+q$, if it satisfies $q=mn$ and $p=m+n$, then the trinomial can be factored as $\left(x+m\right)\left(x+n\right)$. For example: ① $x^{2}+5x+6=\left(x+2\right)\left(x+3\right)$; ② $x^{2}-5x-6=\left(x-6\right)\left(x+1\right)$.<br/>Material $2$: Factoring: $4\left(x+y\right)^{2}+4\left(x+y\right)+1$.<br/>Solution: Consider "$x+y$" as a whole, let $x+y=m$, then the original expression $=4m^{2}+4m+1=\left(2m+1\right)^{2}$. Then, restore "$m$", we get the original expression $=\left(2x+2y+1\right)^{2}$. The above problem-solving method uses the concept of considering a whole, which is a common method in mathematical problem-solving. Please answer the following questions.<br/>$(1)$ According to Material $1$, factorize: $x^{2}-7x+12$.<br/>$(2)$ Combining Material $1$ and Material $2$, complete the following sub-questions:<br/>① Factorize $\left(x-y\right)^{2}+4\left(x-y\right)+3$;<br/>② Factorize $\left(a+b\right)\left(a+b-2\right)-3$.

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