jmc

Let $h$ be the length of the altitude from $A$ in $△ABC$. Then $$2007=\frac{1}{2}\cdot BC\cdot h=\frac{1}{2}\cdot 223\cdot h,$$so $h=18$. Thus $A$ is on one of the lines $y=18$ or $y=-18$.



Line $DE$ has equation $x-y-300=0$. Let $A$ have coordinates $(a,b)$. By the formula for the distance from a point to a line, the distance from $A$ to line $DE$ is $|a-b-300|/\sqrt{2}$. The area of $△ADE$ is $$7002=\frac{1}{2}\cdot \frac{|a-b-300|}{\sqrt{2}}\cdot DE=\frac{1}{2}\cdot \frac{|a\pm 18-300|}{\sqrt{2}}\cdot 9\sqrt{2}.$$Thus $a=\pm 18\pm 1556+300$, and the sum of the four possible values of $a$ is $4\cdot 300=\boxed{1200}$.