The graph of $y=f(x)$, where $f(x)$ is a polynomial of degree $3$, contains points $A(1,2)$, $B(2,5)$, and $C(3,10)$. Lines $AB$, $AC$, and $BC$ intersect the graph again at points $D$, $E$, and $F$, respectively, and the sum of the $x$-coordinates of $D$, $E$, and $F$ is $12$. What is $f(0)$?