AMC 12 2002a Problem 19

The graph of the function $f$ is shown below. How many solutions does the equation $f(f(x))=6$ have?

import graph;
size(200);
defaultpen(fontsize(10pt)+linewidth(.8pt));
dotfactor=4;

pair P1=(-7,-4), P2=(-2,6), P3=(0,0), P4=(1,6), P5=(5,-6);
real[] xticks={-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6};
real[] yticks={-6,-5,-4,-3,-2,-1,1,2,3,4,5,6};

draw(P1--P2--P3--P4--P5);

dot("(-7, -4)",P1);
dot("(-2, 6)",P2,LeftSide);
dot("(1, 6)",P4);
dot("(5, -6)",P5);

xaxis("$x$",-7.5,7,Ticks(xticks),EndArrow(6));
yaxis("$y$",-6.5,7,Ticks(yticks),EndArrow(6));

$\text{(A) }2 \text{(B) }4 \text{(C) }5 \text{(D) }6 \text{(E) }7$