A physics student models the vertical displacement of a bouncing mechanical component with a polynomial $P(t) = at^3 + bt^2 + ct + d$. The component reaches equilibrium (displacement 0) at $t=1$, $t=2$, and $t=3$ seconds. If the initial displacement at $t=0$ is $12$ cm, what is the displacement of the component at $t=4$ seconds?