All pertinent rigid dimensions are specified for the linkage shown. Note that the ground pivot for the input link is at the origin of the coordinate system. The input angle ${}_{in}$ is currently $250$ measured from the $x$ axis as shown. The figure is not exactly to scale, but it is reasonably close for checking purposes. Use the equations developed from the loop closure method to determine the following information.

$($a$)$ On the figure, show the X axis to be used for consistency with the loop closure equations. Show on the figure the angles ${}_2,{}_3,$ and ${}_4.$

$($b$)$ Calculate ${}_3$ and ${}_4$ consistent with the notation of the loop closure equations.

$($c$)$ Using the results of part $($b$)$ with a coordinate orientation change, determine the value of the angle ${}_{\text{o}\text{u}\text{t}}$ as shown on the figure.

$($d$)$ Calculate the absolute location of point P with respect to the coordinate $x-y$ axes shown.