
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 material: Mathematical research has found that common methods of factorization include extracting common factors and using formulas. However, there are many polynomials that cannot be factored using the above methods, such as "$m^{2}-mn+2m-2n$". Careful observation of this expression reveals that the first two terms can have a common factor extracted, as well as the last two terms. After factorizing each part separately, new common factors are generated, and then extracting common factors will complete the factorization of the entire expression. The process is $m^{2}-mn+2m-2n=(m^{2}-mn)+(2m-2n)=m(m-n)+2(m-n)=(m-n)(m+2)$. This method of factorization is called "grouping decomposition". Please solve the following problems inspired by this method:<br/>$(1)$ Factorize: $a^{3}-3a^{2}-4a+12$;<br/>$(2)$ Given $m+n=5$, $m-n=1$, find the value of $m^{2}-n^{2}+2m-2n$.

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