
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
The overall idea is a common method in mathematical problem solving: Below is a process of a student factorizing the polynomial $(x^{2}+2x)(x^{2}+2x+2)+1$. Treating "$x^{2}+2x$" as a whole, let $x^{2}+2x=y$, then the original expression $=y\left(y+2\right)+1=y^{2}+2y+1=\left(y+1\right)^{2}$, then restore "$y$" to get the final result. Solution: Let $x^{2}+2x=y$. The original expression $=y\left(y+2\right)+1=y^{2}+2y+1=\left(y+1\right)^{2}=(x^{2}+2x+1)^{2}$. <br/>Question:<br/>$(1)$ Did the student complete the factorization? If not, please write down the final result directly ______;<br/>$(2)$ Please try to factorize the polynomial $(x^{2}-4x)(x^{2}-4x+8)+16$ following the above method.

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