
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
Material $1$: Using polynomial multiplication, $(x+a)(x+b)=x^{2}+(a+b)x+ab$. By applying this formula from right to left, we can factorize a polynomial in the form of $x^{2}+(a+b)x+ab$ as $(x+a)(x+b)$. The characteristics of the polynomial $x^{2}+(a+b)x+ab$ are that the coefficient of the quadratic term is $1$, the constant term is the product of the two numbers, and the coefficient of the linear term is the sum of these two numbers.

Material $2$: Factorization: $(x+y)^{2}+2(x+y)+1$. Solution: Consider "$x+y$" as a whole, let $x+y=A$, then the original expression $=A^{2}+2A+1=(A+1)^{2}$, then restore "$A$" to get: the original expression $=(x+y+1)^{2$.

The above problem-solving method uses the concept of considering a whole as a single entity, which is a common method in mathematical problem-solving. Please answer the following questions:

$(1)$ Factorize $x^{2}+4x+3$ according to Material $1$.

$(2)$ Factorize $\left(x-y\right)^{2}-10\left(x-y\right)+25$ according to Material $2$.

$(3)$ Combining Material $1$ and Material $2$, factorize $(m^{2}-2m)(m^{2}-2m+4)+3$.

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