$(30$ Points$)$ Consider a shaft subjected to the applied torques at $B$ and $C.$ The right end is clamped, and the left end is connected to the nonlinear torsional spring. The torsional spring exhibits the angle of twist due to the torsional loading, as follows. Express the answer in terms of given variables in the figure.

a$)$ Determine applied torques $(T)$ when the reaction torque at A becomes nonzero. Now, assume that the twist of the spring is larger than ${}^{**}$ and answer the following questions.

b$)$ Determine the reaction torque at $A(T_A)$ and $D(T_D).$

c$)$ Determine the twist angle at $B({}_B)$ and $C({}_C).$