
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 content before solving the problem.  
Example: If $m^{2}+2mn+2n^{2}-6n+9=0$, find the values of $m$ and $n$.  
Solution:  
$\because m^{2}+2mn+2n^{2}-6n+9=0$  
$\therefore m^{2}+2mn+n^{2}+n^{2}-6n+9=0$  
$\therefore \left(m+n\right)^{2}+\left(n-3\right)^{2}=0$  
$\therefore m+n=0$, $n-3=0$  
$\therefore m=-3$, $n=3$  
Question (1): If $x^{2}+2y^{2}-2xy+4y+4=0$, find the value of $x^{y}$.  
(2) Given $a$, $b$, $c$ are the three sides of $\triangle ABC$, satisfying $a^{2}+b^{2}=10a+8b-41$, and $c$ is the longest side of $\triangle ABC$, find the range of values for $c$.

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