
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
Reading material: The textbook mentions that expressions like $a^{2}+2ab+b^{2}$ and $a^{2}-2ab+b^{2}$ are called perfect square trinomials. Some polynomials are not perfect square trinomials. We can add terms to make them perfect square trinomials, then subtract these added terms to keep the value of the entire expression unchanged. This method can also be used to factorize polynomials and solve some extremum problems.  
For example:  
$(1)$ Factorize the expression: $x^{2}-2x-3$.  
$x^{2}-2x-3$  
$=x^{2}-2x+1-1-3$  
$=\left(x-1\right)^{2}-4$  
$=\left(x-1\right)^{2}-2^{2}$  
$=\left(x-1+2\right)\left(x-1-2\right)$  
$=\left(x+1\right)\left(x-3\right)$.  
$(2)$ Find the minimum value of the algebraic expression $x^{2}-2x-3$.  
$x^{2}-2x-3=x^{2}-2x+1-4=\left(x-1\right)^{2}-4$  
$\because \left(x-1\right)^{2}\geqslant 0$  
$\therefore$ When $x=1$, the algebraic expression $x^{2}-2x-3$ has a minimum value of $-4$.  
Based on the above material, solve the following problems:  
$(1)$ Factorize the expression: $x^{2}-6x+8$;  
$(2)$ For what value of $x$ does $x^{2}-6x+8$ have the minimum value? What is the minimum value?

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