Let $~g\left( x \right)=\cos \left( 3x+\frac{\pi }{4} \right)~$ 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 $~{{T}_{3}}\left( x \right)~$ can be written as $~a\sqrt{2}\,$. Determine $~a\,$.