A four-digit number $Y E A R$ is called very good if the system  $$  \begin{aligned}  & Y x+E y+A z+R w=Y \\  & R x+Y y+E z+A w=E \\  & A x+R y+Y z+E w=A \\  & E x+A y+R z+Y w=R  \end{aligned}  $$  of linear equations in the variables $x, y, z$ and $w$ has at least two solutions. Find all very good YEARs in the 21st century and provide the sum of all these years.

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