The Enigma of the Cosmic Journey

A space probe travels along a path described by the function

$�(�)=�{�}^3+�{�}^2+��+�$, where $�$, $�$, $�$, and $�$ are constants. The probe starts its journey at $�=0$ and reaches its furthest point at $�=4$ hours, where it then changes its course. Given that the probe's path reaches a maximum height at $�=2$ hours and the heights at $�=0$, $�=2$, and $�=4$ are $0$, $64$, and $0$ meters respectively, determine the values of $�$, $�$, $�$, and $�$.

Hint: Use the conditions given to set up equations and solve for the unknowns.