
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 and solve the following problems. For a quadratic trinomial like $x^{2}+2ax+a^{2}$, it can be factored into the form of $\left(x+a\right)^{2}$ using the formula method. However, for the quadratic trinomial $x^{2}+2ax-3a^{2}$, the formula cannot be directly applied. In this case, we can add $a^{2}$ to the quadratic trinomial $x^{2}+2ax-3a^{2}$, making it a perfect square with $x^{2}+2ax$, then subtract $a^{2}$. The value of the entire expression remains unchanged. Therefore, we have: $x^{2}+2ax-3a^{2}=(x^{2}+2ax+a^{2})-a^{2}-3a^{2}=\left(x+a\right)^{2}-\left(2a\right)^{2}=\left(x+3a\right)\left(x-a\right)$. This method of first adding an appropriate term to create a perfect square in the expression, then subtracting that term to keep the value unchanged is called "completing the square."

$(1)$ Using "completing the square" method to factorize: $a^{2}-6a+8$.

$(2)$ If $a+b=5$ and $ab=6$, find: ① $a^{2}+b^{2}$; ② $a^{4}+b^{4}$.

$(3)$ Given that $x$ is a real number, compare $x^{2}-4x+5$ with $-x^{2}+4x-4$, and explain the reasoning.

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