
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: Given real numbers $m$ and $n$ satisfying $(2m^{2}+n^{2}+1)(2m^{2}+n^{2}-1)=80$, find the value of $2m^{2}+n^{2}$.  
Solution: Let $2m^{2}+n^{2}=t$, then the original equation becomes $(t+1)(t-1)=80$, which simplifies to $t^{2}-1=80$, i.e., $t^{2}=81$.  
Therefore, $t=\pm 9$.  
Since $2m^{2}+n^{2}\geqslant 0$, we have $2m^{2}+n^{2}=9$.  
The method used above is called "substitution method". Substitution method is the most commonly used method in mathematical learning. In the operation of complex numbers and expressions, if we regard some parts as a whole and replace them with new letters (i.e., substitution), it can simplify complex problems.  
Based on the above reading material, solve the following problems and write down the solution process.  
$(1)$ Given real numbers $x$ and $y$ satisfying $(2x^{2}+2y^{2}+3)(2x^{2}+2y^{2}-3)=27$, find the value of $x^{2}+y^{2}$.  
$(2)$ If the product of four consecutive positive integers is $120$, find these four consecutive positive integers.

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