A vertical curve connects a $+2\%$ initial grade with a $-1\%$ final grade and is designed for $55$ mph$.$ The station of PVI is $233+40$ with elevation $1203$ ft$.$ What is the minimum length of the curve to ensure adequate stopping sight distance for a driver with an eye height of $3$ ft$?$ Design Values $($unless otherwise specified in the problem$)$:

Perception$-$Reaction Time: $2$ s

Coefficient of Friction: $0\mathrm{.}30$

Coefficient of Side Friction: $0\mathrm{.}15$

Max. Superelevation: $8\%$

Max. Grade: $7\%$

Lane Width: $12$ ft

Driver eye height: $3\mathrm{.}5$ ft

Obstruction height: $2$ ft

Headlight height: $2$ ft

Headlight angle: $1$ degree