
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
【Reading Material】  
If $m^{2}+2mn+2n^{2}-4n+4=0$, find the values of $m$ and $n$.  
Solution: From the given condition, we have $(m^{2}+2mn+n^{2})+(n^{2}-4n+4)=0$.  
Therefore, $\left(m+n\right)^{2}+\left(n-2\right)^{2}=0$.  
Solving, we get $\left\{\begin{array}{l}m=-2\\ n=2\end{array}\right.$.  
So, $m=-2$, $n=2$.  
【Problem Resolution】  
$(1)$ Given that the side lengths $a$, $b$, $c$ of $\triangle ABC$ satisfy $a^{2}+b^{2}=12a+8b-52$, and $c$ is the longest side and is even, find the maximum perimeter of $\triangle ABC$;  
$(2)$ For the algebraic expression $x^{2}+4xy+5y^{2}-4y+10$, does it have a maximum or minimum value? What values should $x$ and $y$ take in that case? Find the maximum or minimum value of the expression.

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