Solve the system $$\left\{\begin{array}{l}x^2+y^2-2z^2=2a^2,\\x+y+2z=4(a^2+1),\\z^2-xy=a^2.\end{array}\right.$$ Please provide the sum of the elements for all solutions when $a=1$.

Let's think step by step and output the final answer within \boxed{}.
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