
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 first: We have learned methods of factoring polynomials such as common factor method and formula method. In fact, there are other methods for factoring polynomials, such as grouping method, splitting method, cross multiplication method, etc.  
$(1)$ Grouping method: After appropriately grouping a polynomial, we can factor out common factors or continue factoring using formulas. For example:  
① $ax+by+bx+ay$  
$=(ax+bx)+(ay+by)$  
$=x(a+b)+y(a+b)$  
$=(a+b)(x+y)$  
② $2xy+y^{2}-1+x^{2}$  
$=x^{2}+2xy+y^{2}-1$  
$=(x+y)^{2}-1$  
$=(x+y+1)(x+y-1)$  
$(2)$ Splitting method: By splitting one term of a polynomial into two terms, we can factor out common factors or continue factoring using formulas. For example:  
$x^{2}+2x-3$  
$=x^{2}+2x+1-4$  
$=(x+1)^{2}-2^{2}$  
$=(x+1+2)(x+1-2)$  
$=(x+3)(x-1)$  
Please explore and solve the following problems following the methods above:  
$(1)$ Factorize: $a^{2}-b^{2}+a-b$;  
$(2)$ Factorize: $a^{2}+4ab-5b^{2}$;  
$(3)$ Does the polynomial $x^{2}-6x+1$ have a minimum value? If so, what is the value of $x$ when it reaches the minimum value?

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