
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
By adjusting the algebraic expression to obtain a perfect square trinomial, and then performing relevant operations and problem-solving, this solving method is called the method of adjustment. For example:  
① Factorize the expression using the method of adjustment: $a^{2}+4a+3$.  
Solution: Original expression: $=a^{2}+4a+4-1=\left(a+2\right)^{2}-1=\left(a+2+1\right)\left(a+2-1\right)=\left(a+3\right)\left(a+1\right)$;  
② $M=2a^{2}-4a+6$, find the minimum value of $M$ using the method of adjustment.  
Solution: $M=2a^{2}-4a+6=2(a^{2}-2a+1)+6-2=2\left(a-1\right)^{2}+4$,  
Since $2\left(a-1\right)^{2}\geqslant 0$, therefore $2\left(a-1\right)^{2}+4\geqslant 4$,  
Therefore, when $a=1$, $M$ has a minimum value of $4$.  
Please solve the following problems based on the above information:  
$(1)$ Factorize $x^{2}-4x-12$ using the method of adjustment;  
$(2)$ If $M=4x^{2}+4x-1$, find the minimum value of $M$.

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