Given integer $a = 100$.

Consider points $A(2a,0), B(0,2a), C(a,0), D(0,a)$ and a line $AB$.

Point $E$ is on the line at point $(x,y)$ and the circles intersect again at point $F$.

As Point $E$ moves from $A$ to $B$ let the distance traveled by point $F$ be $d$.

Find $[d]$. Where [.] is the greatest integer function.

Note: A new pair of circles is formed for every location of $E$ using the points $ACE$ and $BDE$. The second point of intersection is labeled $F$