Problem $1(4$ points$)$: Consider the $2-$stage, reverted compound planetary gear train. In the first planetary stage $($gears $2,4,$ and carrier $3),$ gear $2$ is the sun gear, gear $4$ is the planet; in the second planetary stage $($gears $5,6,$ and carrier $3),$ gear $5$ is the planet, and gear $6$ is the sun. The two sets have the same carrier. Gears $4$ and $5$ are on the same shaft.

a$.$ If all the gears have same diametral pitch, find the relationship between the number of teeth on gears $2,4,5,$ and $6.$ Using this, obtain number of teeth on gear $6.$

b$.$ Consider gears $2$ and $4,$ and carrier $3.$ Assume gear $2$ to be fixed. Using the tabular method, determine the relationship between ${}_4$ and ${}_3.$ Also, obtain an expression for the relative angular velocity of gear $4$ with respect to the carrier