As shown below, convex pentagon $ABCDE$ has sides $AB=3$, $BC=4$, $CD=6$, $DE=3$, and $EA=7$. The pentagon is originally positioned in the plane with vertex $A$ at the origin and vertex $B$ on the positive $x$-axis. The pentagon is then rolled clockwise to the right along the $x$-axis. Which side will touch the point $x=2009$ on the $x$-axis?

Enter "AB", "BC", "CD", "DE", or "EA".

[asy] unitsize(3mm); defaultpen(linewidth(.8pt)); dotfactor=4;  pair A=(0,0), Ep=7*dir(105), B=3*dir(0); pair D=Ep+B; pair C=intersectionpoints(Circle(D,6),Circle(B,4))[1]; pair[] ds={A,B,C,D,Ep};  dot(ds); draw(B--C--D--Ep--A); draw((6,6)..(8,4)..(8,3),EndArrow(6)); xaxis("$x$",-8,14,EndArrow(3));  label("$E$",Ep,NW); label("$D$",D,NE); label("$C$",C,E); label("$B$",B,SE); label("$(0,0)=A$",A,SW);  label("$3$",midpoint(A--B),N); label("$4$",midpoint(B--C),NW); label("$6$",midpoint(C--D),NE); label("$3$",midpoint(D--Ep),S); label("$7$",midpoint(Ep--A),W); [/asy]
Side $\overline{AB}$ covers the interval $[0,3],$ then $\overline{BC}$ covers the interval $[3,7],$ then $\overline{CD}$ covers the interval $[7,13],$ then $\overline{DE}$ covers the interval $[13,16],$ then $\overline{EA}$ covers the interval $[16,23],$ and then the process repeats.  The intervals the sides touch repeat with period 23.

Since $2009 = 87 \cdot 23 + 8,$ side $\boxed{\overline{CD}}$ touches the point 2009.