
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
Given the equation: $a^{2}+b^{2}+4a-2b+5=0$, find the values of $a$ and $b$.  
Method:  
1. When we see $a^{2}+4a$, we can think that adding the constant $4$ would make it $a^{2}+4a+4=\left(a+2\right)^{2}$, this process is called "completing the square". Similarly, $b^{2}-2b+1=\left(b-1\right)^{2}$, which perfectly distributes the constant $5$;  
2. Therefore, the original equation can be transformed into $\left(a+2\right)^{2}+\left(b-1\right)^{2}=0$. By the non-negativity of squares, we get $a+2=0$ and $b-1=0$.  
Application:  
(1) If $a^{2}+b^{2}-2a+6b+10=0$, find the value of $a+b$.  
(2) For which values of $a$, $b$, $c$ does the algebraic expression $a^{2}+5b^{2}+c^{2}-2ab-4b+6c+15$ have the minimum value? And what is the minimum value?

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