
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
Xiao Cong and Xiao Ming are students who love to think. During their studies, they discovered that some polynomial multiplications have very characteristic results,  
for example: $(x-2y)(x^2+2xy+4y^2)=x^3-8y^3$, $(x+1)(x^2-x+1)=x^3+1$  
Xiao Cong said: These polynomial multiplications all involve a binomial multiplied by a trinomial on the left side, resulting in a binomial on the right side.  
Xiao Ming said: Yes! And the right side can be seen as the sum (or difference) of the cubes of two terms.  
Xiao Cong said: Moreover, I found that the binomial on the left side somewhat resembles the final result.  
Xiao Ming said: Right, I also noticed that the trinomial on the left seems to be a perfect square form, no, not exactly, because the middle is not twice the product of two terms.  
Xiao Cong said: The sign in the middle of the binomial, the sign in the middle of the trinomial, and the sign in the middle of the result on the right side also seem to be related. ...  
Dear students, can you join their discussion and find the corresponding pattern?  
(1) Please express the pattern you found using letters.  
(2) Please use the above pattern to calculate $(a+2b)(a^2-2ab+4b^2)=$    .  
(3) Please factor the polynomial $a^3-8$ using the above pattern.

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