$(60$ points$)$ For the given $3R$ manipulator in the figure,

a$.$ Justify the degree of freedom.

b$.$ Assign the frames.

c$.$ Fill the DH Parameter table out.

d$.$ Find the forward kinematics.

e$.$ If ${?}^3{P}_{\text{p}\text{i}\text{n}}={\left[\begin{array}{ccc}1& -1& 0\end{array}\right]}^T,$ find ${?}^0{P}_{\text{p}\text{i}\text{n}}.$

f$.$ If ${?}^0{P}_{\text{p}\text{i}\text{n}}={\left[\begin{array}{ccc}0& 1+4L_2+L_1& 1+3L_2\end{array}\right]}^T,$ find the

corresponding values of the joint variables.