$(10$ Points$)$ The two solid steel shafts shown in the figure are coupled together using the meshed gears. Determine the angle of twist of end $A$ of shaft $AB$ when the torque $T=45Nm$ is applied. Take $G=80$GPa. Shaft $AB$ is free to rotate within bearings $E$ and $F,$ whereas shaft $DC$ is fixed at D$.$ Each shaft has a diameter of $20mm.$

Note: Dimensions $L_1$ and $L_2$ are given for each student in the attached table.