
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 $m^{2}-2mn+2n^{2}-8n+16=0$, find the values of $m$ and $n$.  
Solution:  
Since $m^{2}-2mn+2n^{2}-8n+16=0$, then $(m^{2}-2mn+n^{2})+(n^{2}-8n+16)=0$  
Therefore, $\left(m-n\right)^{2}+\left(n-4\right)^{2}=0$  
Hence, $\left(m-n\right)^{2}=0$, $\left(n-4\right)^{2}=0$  
Therefore, $n=4$, $m=4$.  
Based on your observation, investigate the following questions:  
$(1)$ Given $a^{2}+6ab+10b^{2}+2b+1=0$, find the values of $a$ and $b$;  
$(2)$ Given that the three sides $a$, $b$, $c$ of $\triangle ABC$ are all positive integers, and satisfy $2a^{2}+b^{2}-4a-6b+11=0$. Find the perimeter of $\triangle ABC$.

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