Problem $2(6$ points$)$: Consider the planetary gear train. Planets $3$ and $4$ have the same carrier labeled $6.$ All the gears have the same module. Gear $2$ has $12$ teeth, gear $3$ has $16$ teeth, and gear $4$ has $12$ teeth. Ring gear $5$ is fixed.

a$.$ How many teeth does the ring gear $5$ have? Obtain the equation relating the number of teeth on gears $5,2,3,$ and $4.$

b$.$ Focusing on gears $3,$ and $4,$ from the perspective of carrier $6,$ obtain the ratio $\frac{{}_{\frac{4}{6}}}{{}_{\frac{3}{6}}}.$

c$.$ Focus on gears $4,$ and $5,$ and carrier $6.$ If ring gear $5$ is fixed, using the tabular method, relate the angular speeds ${}_4$ and ${}_6.$

d$.$ Lastly, focus on gears $2,$ and $3.$ From the perspective of carrier $6,$ obtain the ratio $\frac{{}_{\frac{2}{6}}}{{}_{\frac{3}{6}}}.$

e$.$ Using results from all parts of the problem, if gear $2$ rotates at a clockwise rate of $1200$ rpm $,$ obtain the angular speeds ${}_6,{}_3,{}_4.$