You have been tasked with designing an equal$-$tangent vertical curve to connect grades of $+1\mathrm{.}0\%$ and $-2\mathrm{.}0\%.$ The design speed for the road is $113$ km$/$h$.$ Assume a deceleration rate of $3\mathrm{.}4$ m$/$s$^2,$ a reaction time of $2\mathrm{.}5$ seconds, and a gravitational constant of $9\mathrm{.}807$ m$/$s$^2.$

a$.$ Calculate the Stopping Sight Distance $($SSD$)$ assuming vehicles travel at the design speed.

b$.$ Using this SSD as the required minimum sight distance, determine the minimum length of the curve. Assume the driver's eye height $($H$1)$ is $1\mathrm{.}08$ meters, and the height of the object on the road $($H$2)$ is $0\mathrm{.}6$ meters.