
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 $m^{2}-2mn+2n^{2}-8n+16=0$, we have $(m^{2}-2mn+n^{2})+(n^{2}-8n+16)=0$

Therefore, $(m-n)^{2}+(n-4)^{2}=0$, which implies $(m-n)^{2}=0$ and $(n-4)^{2}=0$. Thus, $n=4$ and $m=4$.

Based on your observation, explore the following questions:

$(1)$ If $a^{2}+b^{2}-2a+1=0$, then $a=$ ______. $b=$ ______.

$(2)$ Given $x^{2}+2y^{2}-2xy+6y+9=0$, find the value of $x^{y}$.

$(3)$ Given that the lengths of the sides of $\triangle ABC$, $a$, $b$, and $c$, are 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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