a+b+c=0
=>a+b=-c; a+c=-b; b+c=-a
\(a^2=2\left(a+c+2\right)\left(a+b-2\right)\)
=>\(a^2=2\left(-b+2\right)\left(a+b-2\right)\)
=>\(a^2=-2\left(b-2\right)\left(b-2+a\right)\)
=>\(a^2=-2\left(b-2\right)^2+2a\left(b-2\right)\)
=>\(a^2-2a\left(b-2\right)+\left(b-2\right)^2=-\left(b-2\right)^2\)
=>\(\left(a-b+2\right)^2=-\left(b-2\right)^2\)
=>\(\begin{cases}a-b+2=0\\ b-2=0\end{cases}\Rightarrow\begin{cases}b=2\\ a-2+2=0\end{cases}\Rightarrow\begin{cases}b=2\\ a=0\end{cases}\)
c=-(a+b)=-2
\(A=a^{2022}+b^{2022}+c^{2022}\)
\(=0^{2022}+2_{}^{2022}+\left(-2\right)^{2022}=2\cdot2^{2022}=2^{2023}\)
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