Let $~g\left( x \right)=e^{2x+2}~$ and let $~{{T}_{3}}\left( x \right)~$ be the third-degree Taylor polynomial for $~g~$ centered at $~x=0\,$. The sum of the coefficients of $~T_3~$ can be written as $~ae^2$. Determine $~a\,$.
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